\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2021 (2021), No. 69, pp. 1--52.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2021. This work is licensed under a CC BY 4.0 license.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2021/69\hfil Quadratic systems and Darboux invariants]
{Quadratic systems with an invariant algebraic curve of degree 3 and a Darboux invariant}
\author[J. Llibre, R. D. S. Oliveira, C. A. B. Rodrigues \hfil EJDE-2021/69\hfilneg]
{Jaume Llibre, Regilene D. S. Oliveira, Camila A. B. Rodrigues}
\address{Jaume Llibre \newline
Departament de Matematiques,
Universitat Aut\`onoma de Barcelona, 08193 Bellaterra,
Barcelona, Catalonia, Spain}
\email{jllibre@mat.uab.cat}
\address{Regilene Oliveira \newline
Instituto de Ci\^encias Matem\'aticas e Computa\c{c}\~ao,
Universidade de S\~ao Paulo,
Avenida Trabalhador S\~ao-carlense, 400, 13.560-970,
S\~ao Carlos, SP, Brazil}
\email{regilene@icmc.usp.br}
\address{Camila A. B. Rodrigues \newline
Departamento de Matem\'{a}tica,
Universidade Federal de Santa Catarina, 88040-900,
Florian\'opolis, Santa Catarina, Brazil}
\email{c.r.lima@ufsc.br}
\thanks{Submitted September 25, 2020. Published August 16, 2021.}
\subjclass[2010]{34C05, 34A34, 34C23}
\keywords{Quadratic vector fields; algebraic invariant curve; Darboux invariant;
\hfill\break\indent global phase portrait}
\begin{abstract}
Let $QS$ be the class of non-degenerate planar quadratic differential systems
and $QS_3$ its subclass formed by the systems possessing an invariant cubic $f(x,y)=0$.
In this article, using the action of the group of real affine transformations and time
rescaling on $QS$, we obtain all the possible normal forms for the quadratic
systems in $QS_3$. Working with these normal forms we complete the characterization of
the phase portraits in $QS_3$ having a Darboux invariant of the form $f(x, y)e^{st}$,
with $s \in \mathbb R$.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks
\section{Introduction and statements of results}
Even after hundreds of studies on the topology of real planar
quadratic vector fields the complete characterization of their phase
portraits is a quite complex task. This family of systems depends on
twelve parameters but, after affine transformations and time rescaling, we arrive at
families with five parameters, which is still a big number of parameters.
Many subclasses have been considered.
Denote by $\mathbb R[x, y]$ the ring of the real polynomials in the
variables $x$ and $y$. Consider the differential system
in $\mathbb{R}^2$ given by
\begin{equation}\label{eq1}
\dot{x}=P(x,y),\quad \dot{y}=Q(x,y),
\end{equation}
where $P,Q \in \mathbb R[x, y]$. Here the dot denotes derivative with
respect to the \emph{time} $t$ and the degree of system \eqref{eq1} is
$m = \max \{ \deg P, \deg Q \}$.
When $m=2$ we say that system \eqref{eq1} is a \emph{quadratic polynomial differential
system} or simply a \emph{quadratic system}. More
than one thousand papers have been published about quadratic
systems, see for instance \cite{reys} for a bibliographical survey.
The quadratic systems appear in the modeling of many natural
phenomena described in different branches of science, in biological
and physical applications. Besides the applications the quadratic
systems became a matter of interest for the mathematicians. Considering algebraic
invariant curves, some authors have published on the subject, see for
instance \cite{vulpecristina} and \cite{messias_reinol_llibre}.
In the first one the authors studied cubic systems with invariant straight lines of
total multiplicity eight that have three distinct infinite singularities.
The second paper is dedicated to study the normal forms and global phase portraits of
quadratic and cubic integrable systems when they have two nonconcentric circles as
invariant algebraic curves. The global phase portrait of a quadratic system is also
investigated in \cite{pessoa}, where the authors classified all global phase portraits
of Bernoulli quadratic polynomial differential systems in $\mathbb{R}^2$.
In this paper we assume that the polynomials $P$ and $Q$ are coprime, otherwise system
\eqref{eq1} can be reduced to a linear or constant system doing a rescaling of the time
variable.
The first objective of this paper is to characterize all quadratic systems having
invariant cubics. Then using the normal forms obtained, we investigate which systems
have a Darboux invariant either of the form $e^{st} f_1^{\lambda_1}f_2^{\lambda_2}f_3^{\lambda_3}$
if the cubic is the product of three straight lines $f_i=0$ for $i=1,2,3$ or of the form
$e^{st} f_1^{\lambda_1}f_2^{\lambda_2}$ if the cubic is the product of one straight line $f_1=0$
and an irreducible conic $f_2=0$ or of the form $e^{st} f_1^{\lambda_1}$ if $f_1=0$ is an
irreducible cubic.
The paper is organized as follows. In section \ref{statements} we present our main results.
They are divided in two subsections. In section \ref{secbasicresults} we present definitions
and results that are used for proving our main results.
Finally in sections \ref{section3}, \ref{proofteo5} and \ref{secanalysis} we prove the main
results.
\section{Statement of the main results}
\label{statements}
Since the cubic curves can be classified as reducible and irreducible curves (according to
the polynomial defining the curve admits fatorization or not), we split the obtained results
in two subsections. In the first one we consider planar quadratic systems having
irreducible cubics and in the second one, the reducible ones.
\begin{theorem} \label{irreduciblenormalforms}
Each quadratic system admitting an irreducible invariant cubic after an affine change
of coordinates and a rescaling of the time variable can be written as one of the
following systems, where $a,b,c,d$ and $r$ are real numbers,
\begin{itemize}
\item[(i)] $\dot{x}=2 (a x+b y+d x y+c x^2)$,\\
$\dot{y}=3(a y+b x^2+c x y+d y^2)$,
\item[(ii)] $\dot{x}=2(a x+by+(3 b-2 c)xy+a x^2)$,\\
$\dot{y}=2 b x+2 a y+2 c x^2+3 a x y+(9 b-6 c)y^2$,
\item[(iii)] $\dot{x}=2(a x-b y+ (3 b+2 c)xy-a x^2)$,\\
$\dot{y}=2 b x+2 a y+2 c x^2-3 a x y+(9 b+6 c)y^2$,
\item[(iv)] $\dot{x}=2y (a+b x)$, \\
$\dot{y}=a r- 2(a r+a+b r)x+ (3 a+b r+b)x^2+3 b y^2$,
\item[(v)] $\dot{x}=2 y (b+c x)$, \\
$\dot{y}=b+2(b r- c)x+(3 b-c r)x^2+3 c y^2$.
\end{itemize}
\end{theorem}
\begin{theorem}
\label{irreducibletheorem}
Each quadratic system admitting an irreducible invariant cubic having a Darboux invariant
can be written after an affine change of coordinates and a rescaling of the time variable as
\begin{equation}
\label{irreducible}
\dot{x}=x+y, \quad \dot{y}=\frac{3}{2}y+x^2.
\end{equation} After the change of coordinates, $y^2-(2/3) x^3$ is the invariant
algebraic curve and the Darboux invariant is given by
$I_1(t,x,y)=e^{-3 t}\left(\frac{2}{3}x^3+y^2\right) $. The global phase portrait of such
system is given in Figure \ref{fig1}.
\end{theorem}
\begin{figure}[http]
\centering
\includegraphics[scale=0.7]{fig1}
\caption{Phase portrait of system \eqref{irreducible}.}
\label{fig1}
\end{figure}
Theorems \ref{irreduciblenormalforms} and \ref{irreducibletheorem} are proved in
section \ref{section3}.
\subsection{Reducible invariant cubics}
Each reducible cubic can be written as the product of two polynomials one of degree two
and the other of degree one. The conics can be classified in ellipses (E),
complex ellipses (CE), hyperbolas (H), parabolas (P), two real straight lines intersecting
in a point, two real parallel straight lines (PL), one double invariant real straight line (DL),
two complex straight lines intersecting in a real point (p), and two complex parallel straight
lines (CL). So the normal forms of the reducible cubics, except to an affine transformation, are
\begin{itemize}
\item [(E)] $(x^2+y^2-1)(ax+by+c)=0,$
\item[(CE)] $(x^2+y^2+1)(ax+by+c)=0,$
\item[(H)] $(x^2-y^2-1)(ax+by+c)=0,$
\item[(P)] $(y-x^2)(ax+by+c)=0, $
\item[(LV)] $xy(ax+by+c)=0, $
\item[(PL)] $(x^2-1)(ax+by+c) =0,$
\item[(DL)] $x^2(ax+by+c)=0, $
\item[(CL)] $(x^2+1)(ax+by+c)=0, $
\item[(p)] $(x^2+y^2)(ax+by+c)=0.$
\end{itemize}
We shall say that a quadratic system is of type (E) if it has a real ellipse and a straight
line as invariant irreducible algebraic curves; of type (CE) if it has a complex ellipse
and a straight line as invariant irreducible algebraic curves, and respectively with all
the nine types of conics described above.
The first result of this paper classifies the quadratic systems having a reducible invariant
cubic.
\begin{theorem}
\label{teo1}
If a quadratic system \eqref{eq1} has a reducible invariant cubic then it can be written,
after an affine change of coordinates, into one of the following forms
\begin{itemize}
\item[(CE)] $\dot{x}=-(x^2+y^2+1)-2 \alpha_1 y (y+a x+c)$,\\
$\dot{y}=a (x^2+y^2+1)+2 \alpha_1 x (y+a x+c)$,
\item[(E.1)] $\dot{x}=-(x^2+y^2-1)-2 \alpha_1 y (y+a x+c)$,\\
$\dot{y}=a (x^2+y^2-1)+2 \alpha_1 x (y+a x+c)$,
\item[(E.2)] $\dot{x}=(\beta_1/2)(x^2+y^2-1)-y(\beta_2y -\alpha_2 x+ c \beta_2)$,\\
$\dot{y}=(y+c)(\alpha_2 y- \beta_2 c x+\alpha_2)$, where $\alpha_2(c+1)=0$,
\item[(H.1)] $\dot{x}=(\beta_1/2)(x^2-y^2-1) +\beta_2 y(y+c)$,\\
$\dot{y}=\beta_2 y(y+c)$,
\item[(H.2)] $\dot{x}=(x+c)(\alpha_2 x+ \gamma_2 y+\alpha_2)$,\\
$\dot{y}=-(\gamma_1/2)(x^2-y^2-1)+x(\gamma_2 x +\alpha_2 y+ c \gamma_2)$,
where $\alpha_2(c+1)=0$,
\item[(H.3)] $\dot{x}=(A/2) (x^2-y^2-1)-y (\alpha -c\beta+x (\beta -c\alpha))
-y^2 (\gamma -c\alpha)$,\\
$\dot{y}=(A/2)(x^2-y^2-1)-x (\alpha -c\beta+\beta x+y (\gamma -c\alpha))+c\alpha (y^2+1)$,
\\
where $c(\gamma+\beta)=0$,
\item[(H.4)] $\dot{x}=(A/2)(x^2-y^2-1)+y(a \alpha-\beta\sqrt{d}
+x (a \beta - \alpha \sqrt{d}) +\beta y)$,\\
$\dot{y}=(-Aa/2)(x^2-y^2-1)+x(a \alpha- \beta \sqrt{d}+a \beta x+ \beta y)-\alpha\sqrt{d}(y^2+1)$,\\
where $d=a^2-1$,
\item[(H.5)] $\dot{x}=-(x^2-y^2-1)+2 \alpha_1 y (y+a x+c)$, \\
$\dot{y}=a(x^2-y^2-1)+2 \alpha_1 x (y+a x+c)$, where $c^2\neq a^2-1$,
\item[(P.1)] $\dot{x}=x (\alpha_2+\beta_2 x+\gamma_2 y)$, \\
$\dot{y}=\alpha_1( y-x^2)+ 2 \alpha_2 x^2+2y(\beta_2 x+\gamma_2 y)$,
\item[(P.2)] $\dot{x}=-\beta_1(y-x^2)+y(\beta_2+\gamma_2 x)+(\alpha_2+\gamma_2 c)x+c \beta_2$, \\
$\dot{y}=2(y+c) (\alpha_2+\beta_2 x+\gamma_2 y)$, with $c \alpha_2=0$,
\item[(P.3)] $\dot{x}=-(y-x^2)-\alpha(y+a x+c)$,\\
$\dot{y}=a (y-x^2)-2 \alpha x (y+a x+c)$, with $c\neq a^2/4$,
\item[(LV.1)] $\dot{x}=x(\alpha +r y+\beta x)$,\\
$\dot{y}=y (\alpha + (r-q+\beta) y+q x)$,
\item[(LV.2)] $\dot{x}=x (p+q x+r y)$,\\
$\dot{y}=y(y+c)$, with $c(c+1)=0$,
\item[(LV.3)] $\dot{x}=-x (y+\alpha(y+a x+c))$,\\
$\dot{y}=y (a x+\beta(y+a x+c))$, with $a\,c \neq 0$,
\item[(RPL)] $\dot{x}=x^2-1$,\\
$\dot{y}=y (\alpha +\beta x+\gamma y)$,
\item[(DL)] $\dot{x}=x^2$,\\
$\dot{y}=y (\alpha +\beta x+\gamma y)$,
\item[(CPL)] $\dot{x}=x^2+1$, \\
$\dot{y}=y (\alpha +\beta x+\gamma y)$,
\item[(p.1)] $\dot{x}=(\beta/2) (x^2+y^2)-\beta_3 y^2+x (\alpha_3+\gamma_3 y)$, \\
$\dot{y}=y (\alpha_3+\beta_3 x+\gamma_3 y)$,
\item[(p.2)] $\dot{x}=-(x^2+y^2)+(\beta x-\alpha y) (y+a x+c)$,\\
$\dot{y}=a (x^2+y^2)+(\beta y+\alpha x) (y+a x+c)$, with $c \neq 0$,
\end{itemize}
where $a, c, d, A, p, q, r, \alpha, \beta, \gamma, \alpha_1, \beta_1, \gamma_1, \alpha_2, \beta_2, \gamma_2, \alpha_3, \beta_3$ and $\gamma_3$
are the parameters of the system.
\end{theorem}
\begin{theorem}
\label{phaseportraits}
The global phase portrait in the Poincar\'e disc of each quadratic differential system
admitting a reducible invariant cubic $f(x,y)=0$ and having a Darboux invariant of
the form $e^{-st} f(x,y)$ is topologically equivalent to one of the phase portraits
presented in Figures \ref{pp_1}-\ref{pp_6}. Their normal forms according to
Theorem \ref{teo1} is labelled in the corresponding figure.
\end{theorem}
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig2} % Phase_portraits_1.eps
\caption{Phase portraits of systems of type $(E)$ and $(H)$ when they have a Darboux invariant. Phase portraits EL.2.1 and EL.2.2 correspond to system (E.2); HL.2.1--HL.2.3 correspond to system (H.2); HL.3.1--HL.$3.9$ correspond to system (H.3). The dashed lines denote a curve filled of singular points.}
\label{pp_1}
\end{figure}
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig3} %Phase_portraits_2.eps
\caption{Phase portraits of systems of type $(P)$ when they have a Darboux invariant. Phase portraits PL.1.1--PL.1.24 correspond to system (P.1). }
\label{pp_2}
\end{figure}
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig4} % Phase_portraits_3.eps
\caption{Phase portraits of systems of type $(P)$ when they have a Darboux invariant. Phase portraits PL.1.25--PL.1.30 correspond to system (P.1); PL.2.1--PL.$2.11$ correspond to system (P.2). The dashed lines denote a curve filled of singular points.}
\label{pp_3}
\end{figure}
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig5} % Phase_portraits_4.eps}
\caption{Phase portraits of systems of type $(LV)$ when they have a Darboux invariant. Phase portraits LVL.1.1--LVL.1.6 correspond to system (LV.1); LVL.2.1--LVL.$2.17$ correspond to system (LV.2). The dashed lines denote a curve filled of singular points.}
\label{pp_4}
\end{figure}
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig6} % Phase_portraits_5.eps
\caption{Phase portraits of systems of type (RPL) and (DL) when they have a Darboux invariant. Phase portraits RPL.$1$--RPL.$17$ correspond to system (RPL); DL.$1$--DL.$3$ correspond to system (DL). The dashed lines denote a curve filled of singular points.}
\label{pp_5}
\end{figure}
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig7} % Phase_portraits_6.eps}
\caption{Phase portraits of systems of type (CPL) and (p) when they have a Darboux invariant. Phase portraits CPL.$1$--CPL.$7$ correspond to system (CPL); p.1.2 and p.1.2 correspond to system (p.1); p.2.1 and p.2.2 correspond to system (p.2). The dashed lines denote a curve filled of singular points.}
\label{pp_6}
\end{figure}
\begin{theorem} \label{theo2}
Systems of type {\rm (CE), (E.1), (H.1), (H.5), (P.3)} do not admit Darboux invariants
of the form $e^{-st} f(x,y)$.
\end{theorem}
\section{Preliminary and basic results} \label{secbasicresults}
The goal of this section is introduce some definitions and results
which are used in next sections for the study of the Darboux
invariants and to obtain the global phase portrait of the systems of
Theorems \ref{irreducibletheorem} and \ref{teo1}.
\subsection{Invariants} %\label{secinvariant}
A nonconstant $C^1$ function $H : U = \mathbb R$, defined in the open and dense set
$U \subset \mathbb R^2$, is a \emph{first integral}
of system \eqref{eq1} on $U$ if $H(x(t), y(t))$ is constant for all
of the values of $t$ for which $(x(t), y(t))$ is a solution of
system \eqref{eq1} contained in $U$. In other words, $H$ is a first
integral of system \eqref{eq1} if and only if
\begin{equation*}\label{eq2}
P \frac{\partial H}{\partial x} + Q \frac{\partial H}{\partial y} =
0, \quad \text{for all }(x,y)\in U.
\end{equation*}
An \emph{invariant} of system \eqref{eq1} on the open subset $U$ of
$\mathbb R^2$ is a nonconstant $C^1$ function $I$ in the
variables $x, y$ and $t$ such that $I (x(t), y(t), t)$ is constant
on all solution curves $(x(t), y(t))$ of system \eqref{eq1} contained in
$U$, i.e.
\begin{equation} \label{eq4}
\frac{\partial I}{\partial x} P + \frac{\partial I}{\partial y} Q +
\frac{\partial I}{\partial t}=0,\quad \text{for all }(x,y)\in U.
\end{equation}
On the other hand, given $f \in \mathbb C[x, y]$, we say that the
curve $f(x,y)=0$ is an \emph{invariant algebraic curve} of system
\eqref{eq1} if there exists $K \in \mathbb C[x, y]$ such that
\begin{equation}\label{algebraicinv}
P \frac{\partial f}{\partial x} + Q \frac{\partial f}{\partial y} =
K f.
\end{equation}
The polynomial $K$ is called the \emph{cofactor} of the invariant
algebraic curve $f=0$. When $K=0$, $f$ is a polynomial first
integral. Note that if a real polynomial differential system has a complex
invariant algebraic curve then it has also its conjugate. It is
important to consider the complex invariant algebraic curves of the
real systems because sometimes these force the real integrability
of the system, for more details see Chapter 8 of \cite{Llibre}, or
the subsection \ref{secDarbouxinvariant}.
Let $f,g \in \mathbb C[x,y]$ and assume that $f$ and $g$ are
relatively prime in the ring $\mathbb C[x,y]$, or that $g=1$. Then
the function $\exp(f/g)$ is called a \emph{exponential factor} of
system \eqref{eq1} if for some polynomial $L \in \mathbb C[x,y]$ of
degree at most $m-1$ we have
\begin{equation*} \label{eq3}
P \frac{\partial \exp(f/g)}{\partial x} + Q \frac{\partial \exp
(f/g)}{\partial y} =L \exp(f/g).
\end{equation*}
As previously we say that $L$ is the \emph{cofactor} of the exponential
factor $\exp{(f/g)}$. We observe that in the definition of
exponential factor $\exp(f/g)$ if $f,g \in \mathbb C[x,y]$ then the
exponential factor is a complex function. Again when we look for a
complex exponential factor of a real polynomial system we are
thinking the real polynomial system as a complex polynomial system.
\subsection{Darboux invariants} \label{secDarbouxinvariant}
An invariant $I$ is called a \emph{Darboux invariant} if it can be
written into the form
\begin{equation*} \label{darboux}
I(t,x,y)=f_1^{\lambda_1} \cdots f_p^{\lambda_p} F_1^{\mu_1}\cdots
F_q^{\mu_q} e^{s\,t},
\end{equation*}
where $f_i=0$ are invariant algebraic curves of system \eqref{eq1}
for $i=1,\ldots p$, and $F_j$ are exponential factors of system \eqref{eq1}
for $j= 1, \ldots, q$, $\lambda_i, \mu_j \in \mathbb C$ and $s \in
\mathbb R\setminus \{0\}$.
Observe that, if among the invariant algebraic curves a complex
conjugate pair $f = \operatorname{Re}(f)+ \operatorname{Im}(f) i=0$ and $\bar{f}=\operatorname{Re}(f)-
\operatorname{Im}(f) i=0$ occurs, then the Darboux invariant has a factor of the
form $f^{\lambda} \bar{f}^{\bar \lambda}$, which is the real
multi-valued function
$$
\Big(\big(\operatorname{Re}(f)\big)^2 +\big(\operatorname{Im}(f)\big)^2
\Big)^{\operatorname{Re}(\lambda)}
e^{-2 \operatorname{Im}(\lambda) \arctan(\operatorname{Im}(f)/\operatorname{Re}(f))}.
$$
So, if system \eqref{eq1} is real then
the Darboux invariant is also real, independently of the fact of
having complex invariant curves or complex exponential factors.
The next result is proved in \cite[Proposition 8.4]{Llibre}.
\begin{proposition}
\label{prop10}
Suppose that $f \in \mathbb C[x,y]$ and let $f=f_1^{n_1}\ldots f_r^{n_r}$ be its
factorization into irreducible factors over $\mathbb C[x,y]$.
Then for a polynomial differential system \eqref{eq1}, $f=0$ is an invariant algebraic
curve with cofactor $k_f$ if and only if $f_i=0$ is an invariant algebraic curve for each
$i=1,\ldots,r$ with cofactor $k_{f_i}$. Moreover $k_f=n_1 k_{f_1}+\ldots+n_r k_{f_r}$.
\end{proposition}
The next result, proved in \cite{llibre-00}, explain how to obtain a Darboux invariant
using algebraic invariant curves of a polynomial differential system.
\begin{proposition}\label{prop3}
Suppose that a polynomial system \eqref{eq1} of degree $m$ admits
$p$ invariant algebraic curves $f_i = 0$ with cofactors $k_i$ for
$i = 1, \dots ,p$, $q$ exponential factors $\exp(g_j/h_j)$ with
cofactors $L_j$ for $j = 1, \dots ,q$, then, if there exist
$\lambda_i$ and $\mu_j \in \mathbb C$ not all zero such that
\begin{equation}
\label{igualdade de darboux}
\sum_{i=1}^p \lambda_i k_i + \sum_{j=1}^q \mu_j
L_j=-s,
\end{equation}
for some $s \in \mathbb R \backslash \{0\}$, then substituting
$f_i^{\lambda_i}$ by $|f_i|^{\lambda_i}$ if $\lambda_i \in \mathbb
R$, the real (multi-valued) function
$$
f_1^{\lambda_1} \ldots
f_p^{\lambda_p} \Big( \exp\Big(\frac{g_1}{h_1} \Big)\Big)^{\mu_1} \ldots
\Big(\exp\Big( \frac{g_q}{h_q}\Big)\Big)^{\mu_q} e^{st}
$$
is a Darboux invariant of system \eqref{eq1}.
\end{proposition}
The search of first integrals is a classic tool in order
to describe phase portraits of a $2$--dimensional differential
system. As usual the \emph{phase portrait} of a system is the
decomposition of the domain of definition of this system as union of
all its orbits.
It is well known that the existence of a first integral or an a invariant for a planar
differential system allow to draw its phase portrait. Here we investigate the existence
of invariants of the form $f(x,y)e^{st}$, called Darboux invariants,
see section \ref{secDarbouxinvariant} for details. Such invariants describe the
asymptotic behavior of the solutions of the system. Indeed let $\phi_p(t)$ be the
solution of system \eqref{eq1} passing
through the point $p \in \mathbb R^2$, defined on its maximal
interval $(\alpha_p, \omega_p)$ such that $\phi_p(0)=p$. If
$\omega_p = \infty$ we define the \emph{$\omega$-limit set} of $p$ as
\begin{equation*}
\omega(p)=\{q \in \mathbb R^2: \exists \{t_n\} \mbox{
with } t_n = \infty \mbox{ and } \phi_p(t_n)= q \mbox{ when } n
= \infty \}.
\end{equation*}
In the same way, if $\alpha_p = - \infty$ we define the \emph{$\alpha$-limit set} of $p$
as
\begin{equation*}
\alpha(p)=\{q \in \mathbb R^2:\exists \{t_n\} \mbox{
with } t_n = - \infty \mbox{ and } \phi_p(t_n)= q \mbox{ when }
n = \infty \}.
\end{equation*}
For more details on the $\omega$-- and $\alpha$--limit sets see for
instance \cite[section 1.4]{Llibre}.
The existence of a Darboux invariant of system \eqref{eq1} provides information about
the $\omega$-- and $\alpha$--limit sets of all orbits of system \eqref{eq1}. More
precisely, we have the following result, where the definitions of
Poincar\'e compactification and Poincar\'e disc are given in subsection
\ref{secpoincare}. Its proof can be found in \cite{llibre-oliveira}.
\begin{proposition}\label{limitset}
Let $I(t,x,y)=f(x,y) e^{st}$ be a Darboux invariant of system
\eqref{eq1}. Let $p \in \mathbb R^2$ and $\phi_p(t)$ be the solution of
system \eqref{eq1} with maximal interval $(\alpha_p, \omega_p)$ such
that $\phi_p(0)=p$.
Assume $s>0$. Then if $\omega_p= \infty$ we have that $\omega(p)$ is
contained in the closure of $\{f(x,y)=0\}$ inside the Poincar\'e disc,
and if $\alpha_p=-\infty$ we have that $\alpha(p)$ is contained in $\mathbb S^1$,
i.e. at infinity. When $s<0$ we interchange the roles of
$\omega(p)$ and $\alpha(p)$ with respect to $s>0$.
\end{proposition}
\section{Poincar\'e compactification}\label{secpoincare}
Let ${\mathcal X}= P(x,y) \frac{\partial}{\partial x} + Q(x,y) \frac{
\partial}{\partial y}$
be the planar polynomial vector field of degree $m$ associated to
the polynomial differential system \eqref{eq1}. The Poincar\'e
compactified vector field $\pi({\mathcal X})$ corresponding to
${\mathcal X}$ is an analytic vector field induced on $\mathbb S^2$
as follows (for more details, see \cite{Llibre}).
Let $\mathbb S^2 = \{y=(y_1,y_2,y_3) \in \mathbb{R}^3: y_1^2 + y_2^2 + y_3^2 = 1\}$ and
$T_y \mathbb S^2$ be the tangent plane to $\mathbb
S^2$ at point $y$. We identify $\mathbb R^2$ with
$T_{(0,0,1)}\mathbb S^2$ and we consider the central projection $f:
T_{(0,0,1)}\mathbb S^2 \to \mathbb S^2$. The map $f$ defines two
copies of ${\mathcal X}$ on $\mathbb S^2$, one in the southern
hemisphere and the other in the northern hemisphere. Denote by
${\mathcal X}'$ the vector field $D(f \circ {\mathcal X})$ defined
on $\mathbb S^2\setminus \mathbb S^1$, where
$\mathbb S^1 = \{y \in \mathbb S^2: y_3 = 0\}$ is identified with the infinity
of $\mathbb R^2$.
For extending ${\mathcal X}'$ to a vector field on $\mathbb S^2$,
including $\mathbb S^1$, ${\mathcal X}$ must satisfy convenient
conditions. Since the degree of $\mathcal{X}$ is $m$, $\pi({\mathcal X})$ is
the unique analytic extension of $y_3^{m-1}\mathcal{X}'$ to $\mathbb S^2$. On
$\mathbb S^2 \setminus \mathbb S^1$ there are two symmetric copies of
${\mathcal X}$, and once we know the behavior of $\pi(\mathcal{X})$ near
$\mathbb S^1$, we know the behavior of ${\mathcal X}$ in a
neighborhood of the infinity. The Poincar\'e compactification has the
property that $\mathbb S^1$ is invariant under the flow of
$\pi({\mathcal X})$. The projection of the closed northern
hemisphere of $\mathbb S^2$ on $y_3=0$ under $(y_1,y_2,y_3) \mapsto
(y_1,y_2)$ is called the \emph{Poincar\'e disc}, and its boundary is
$\mathbb S^1$.
Two polynomial vector fields ${\mathcal X}$ and ${\mathcal Y}$ on
$\mathbb R^2$ are topologically equivalent if there exists a
homeomorphism on $\mathbb S^2$ preserving the infinity $\mathbb S^1$
carrying orbits of the flow induced by $\pi({\mathcal X})$ into
orbits of the flow induced by $\pi({\mathcal Y})$ preserving or not
the orientation of all the orbits.
As $\mathbb S^2$ is a differentiable manifold, in order to compute
the explicit expression of $\pi({\mathcal X})$, we consider six
local charts $U_i = \{y \in \mathbb S^2; \ y_i >0\}$ and $V_i = \{y
\in \mathbb S^2; \ y_i <0\}$, where $i=1,2,3$, and the
diffeomorphisms $F_i: U_i \to \mathbb R^2$ and $G_i: V_i \to \mathbb
R^2$, for $i=1,2,3$, which are the inverses of the central
projections from the tangent planes at the points $(1,0,0)$,
$(-1,0,0)$, $(0,1,0)$, $(0,-1,0)$, $(0,0,1)$ and $(0,0,-1)$,
respectively. We denote by $z=(u,v)$ the value of $F_i(y)$ and
$G_i(y)$, for any $i=1,2,3$, therefore $z$ means different things
depending on the local charts where we are working. So after some
computations $\pi({\mathcal X})$ is given by:
% page 8
\begin{gather*}
v^m \Delta(z) \Big(Q\big(\frac{1}{v},\frac{u}{v}\big)-u
P\big(\frac{1}{v},\frac{u}{v}\big), -v
P\big(\frac{1}{v},\frac{u}{v}\big)\Big) \quad \text{in } U_1, \\
v^m \Delta(z) \Big(P\big(\frac{u}{v},\frac{1}{v}\big)-u
Q\big(\frac{u}{v},\frac{1}{v}\big), -v
Q\big(\frac{u}{v},\frac{1}{v}\big)\Big) \quad \text{in } U_2, \\
\Delta(z) (P(u,v),Q(u,v)) \quad \text{in } U_3,
\end{gather*}
where $\Delta(z) = (u^2+v^2+1)^{-(m-1)/2}$. The expressions for
$V_i$'s are the same as that for $U_i$'s but multiplied by the
factor $(-1)^{m-1}$. In these coordinates $v=0$ always denote the
points of the infinity $\mathbb S^1$.
\subsection{Irreducible invariant cubics}
The next results characterize all irreducible cubics, their proofs can be found in \cite{Bix}.
\begin{proposition}[{\cite[Theorem 8.3]{Bix}}]
\label{bix1}
A cubic is non-singular and irreducible and has a flex (a generalized inflection point)
if and only if it can be transformed with affine transformations into either
\[
y^2=x(x-1)(x-r) \quad \text{with } r>1,
\]
or
\[y^2=x(x^2+sx+1) \quad \text{with } -2~~k-1$ then
$ X=Y\big(\prod_{i=1}^{q} f_i \big)+ \sum_{i=1}^{q}\big(\prod_{j=1,j \neq i}^{q} f_j
\big) X_{f_i}$,
where $ X_{f_i}=(-\partial f_i/\partial y, \partial f_i/\partial x)$ is a Hamiltonian vector field, the $h_i$
are polynomials of degree $\leq m-k+1$ and $Y$ is a polynomial vector field of degree
less than or equal $m-k$.
\item [(b)] If $m=k-1$ then
$ %\label{vectorfieldclassif2}
X= \sum_{i=1}^{q}\alpha_i\big(\prod_{j=1,j \neq i}^{q} f_j \big) X_{f_i}$,
where $\alpha_i \in \mathbb C$. In this case a Darboux first integral exists.
\item[(c)] If $m1$ or
$f(x,y)=y^2-x(x^2+sx+1)$ with $-2~~~~1$ only one solution can hold
$a_{00}= a_{02}= a_{10}= a_{20}=b_{01}=b_{11}=0$,
$b_{00}= a_{01}r/2$, $b_{02}= 3a_{11}/2$, $b_{10}= -a_{01} (r+1)-a_{11} r$,
$b_{20}= (3 a_{01}+a_{11} r+a_{11})/2$.
It corresponds to system (iv) of Theorem \ref{irreduciblenormalforms}.
For $f(x,y)=y^2-x(x^2+sx+1)$ we obtain only one solution corresponding to system $(v)$
of the theorem.
\end{proof}
Using the normal forms described in Theorem \ref{irreduciblenormalforms} we investigate
when these systems admit a Darboux invariant of the form $e^{st}f(x,y)$.
\begin{proof}[Proof of Theorem \ref{irreducibletheorem}]
First of all, it is easy to see that the cofactor $K$ of $f$ in systems (ii)--(v)
of Theorem \ref{irreduciblenormalforms} has no constant terms.
Then equation \eqref{igualdade de darboux} becomes $\lambda K+s=0$ which never holds if
$s \in \mathbb{R} \setminus \{0\}$ and $\lambda \in \mathbb C \setminus \{0\}$.
Therefore we conclude that systems (ii)--(v) do not admit a Darboux invariant of the
form $e^{st}f(x,y)$.
Now considering system (i) of Theorem \ref{irreduciblenormalforms}. $f(x,y)=y^2-x^3=0$
is an invariant curve of system (i) with cofactor $K=6 (a+c x+d y)$. In this case the
solution of \eqref{igualdade de darboux} is $c = 0, d = 0, s = -6 a \lambda$.
Taking $\lambda=-s/(6a)$ we obtain the system
\begin{equation*}
\dot{x}= 2(ax+by), \quad \dot{y}= 3(a y+b x^2),
\end{equation*}
with Darboux invariant $I(t,x,y)=e^{-6 a t} (y^2-x^3)$.
The normal form described in Theorem \ref{irreducibletheorem} is obtained doing the
following change of coordinates and rescaling of the time
$$
x = \frac{2 a^2}{3 b^2} X, \quad y = \frac{2 a^3}{3 b^3} Y, \quad
t = \frac{1}{2 a} T.
$$
In this case, the Darboux invariant is written as
$I_1(t,x,y)=e^{-3 t}\big(\frac{2}{3}x^3+y^2\big) $.
Now it remains to study the phase portrait of system \eqref{irreducible}.
This system has two singular points, namely $z_1=(0,0)$ hyperbolic unstable node,
and $z_2=(3/2, -3/2)$ a hyperbolic saddle. Applying the Poincar\'e compactification,
in the local chart $U_1$ the compactified system has no singular points.
However in the local chart $U_2$ the origin $(0,0)$ is a nilpotent singularity.
With the notation of Theorem 3.5 of \cite{Llibre} the compactified system has
$F(u)=-u^5-(3/2)u^6 $ and $G(u)=-4u^2-(7/2)u^3$. Hence the origin of $U_2$ is a
nilpotent stable node. By the previous statements it follows that the phase portrait
of system \eqref{irreducible} is the one illustrated in Figure \ref{fig1}.
\end{proof}
\section{Proof of Theorem \ref{teo1}}
\label{proofteo5}
The proof is divided in nine parts, according to the type of the conic in the reducible cubic.
\subsection{Systems of type (E)}
If system \eqref{eq1} has an invariant cubic of the form $f(x,y)=f_1(x,y) f_2(x,y)$ with
$f_1=x^2+y^2-1$ and $f_2=a x+b y+c$, then applying a rotation we can assume $b=1$.
Therefore it follows from Proposition \ref{prop10} that $f_j$ is an invariant curve with
cofactor $k_j =\alpha_j + \beta_j x + \gamma_j y$, $j=1,2$. Now we consider two possibilities:
$a=0$ and $a\neq0$.
If $a=0$ then using equation \eqref{algebraicinv} we have $Q = k_2 f_2$ and
$P=(k_1 f_1-2 y k_2f_2)/(2x)$. As $P$ is a polynomial the parameters of the system must
satisfy on the of following conditions
\begin{gather*}
s_1=\{c= -1, \, \alpha_1= 0,\,\gamma_1= 2 \alpha_2,\,\gamma_2= \alpha_2\},\\
s_2=\{ c= 1,\,\alpha_1= 0,\,\gamma_1=- 2 \alpha_2,\,\gamma_2= -\alpha_2\},\\
s_3=\{\alpha_1= 0,\,\gamma_1= 0,\,\gamma_2= 0\}.
\end{gather*}
Moreover the solutions $s_1$ and $s_2$ provide equivalent systems, and we can
summarize the solutions $s_1$ and $s_3$ writing the system
\begin{equation}
\label{proofE1}
\begin{aligned}
\dot{x}=& (\beta_1/2)(x^2+y^2-1)-y(\beta_2 y -\alpha_2 x+ c \beta_2),\\
\dot{y}=&(y+c)(\alpha_2 y+ \beta_2 c x+\alpha_2),
\end{aligned}
\end{equation}
with $\alpha_2(c+1)=0$. This is exactly system (E.2) of Theorem \ref{teo1}.
When $a \neq0$ we check when the hypotheses of Theorem \ref{CTheo} are satisfied.
Clearly $f_1$ and $f_2$ satisfies (i), (ii) and (iv). Condition (iii) is not satisfied when
$c^2=a^2+1$ because the line $f_2 = 0$ is tangent to the real ellipse $f_1=0$.
Indeed if the straight line $f_2=y+ax+c=0$ is tangent to the real ellipse $f_1=x^2+y^2-1=0$
at the point $(x_0,y_0)$, then their gradients are parallel in such point, what means that
$x_0 -a y_0=0$. Replacing $y_0=x_0/a$ in the ellipse we conclude that $x_0=\pm a/\sqrt{a^2+1}$.
From $f_2=0$ we obtain $c=\mp\sqrt{a^2+1}$. Therefore the condition for the tangency is
$c^2=a^2+1$. By a rotation we obtain $f_2=y-1$. Again we are in system \eqref{proofE1} with $c=-1$.
Now assuming $c^2 \neq a^2+1$, by Theorem \ref{CTheo} the considered system is given by
\begin{equation}\label{proofE2}
\begin{aligned}
\dot{x}&=-\alpha_2(x^2+y^2-1)-2 \alpha_1 y (y+a x+c),\\
\dot{y}&= a \alpha_2 (x^2+y^2-1)+2 \alpha_1 x (y+a x+c),
\end{aligned}
\end{equation}
where $\alpha_1, \,\alpha_2 \in \mathbb{C}$ and $a, c \in \mathbb{R}$. As we are looking for a
real system, then $\alpha_1, \, \alpha_2 \in \mathbb{R}$, and doing a rescaling of the time we can
assume $\alpha_2=1$. Note that system \eqref{proofE2} is exactly system (E.1)
of Theorem \ref{teo1}.
\subsection{Systems of type (CE)} In this case we can follow the same steps applied previously.
If system \eqref{eq1} has an invariant cubic of the form $f=f_1 f_2$ with $f_1=x^2+y^2+1$
and $f_2=a x+b y+c$ we suppose, without loss of generality, $b=1$.
Since the coefficients $a, b$ and $c$ are real numbers the straight line $f_2=0$ cannot be
tangent to the complex ellipse $f_1=0$. So we obtain
\begin{equation} \label{proofCE2}
\begin{aligned}
\dot{x}&=-\alpha_2 (x^2+y^2+1)-2 \alpha_1 y (y+a x+c),\\
\dot{y}&= a \,\alpha_2 (x^2+y^2+1)+2 \alpha_1 x (y+a x+c),
\end{aligned}
\end{equation}
where $\alpha_1, \alpha_2 \in \mathbb{C}$ and $a, c \in \mathbb{R}$.
Applying a rescaling we have $\alpha_2=1$ in \eqref{proofCE2}, and we obtain the normal form
for the systems of type $(CE)$.
\subsection{Systems of type (H)}
Let $f_1=x^2-y^2-1$ and $f_2= a x + by+c$ be two real algebraic invariant curves of
system \eqref{eq1}, so $a^2+b^2 \neq 0$. Proceeding as before if $a=0$ we can assume $b=1$
and the system can be written in the form
\begin{equation*} %\label{proofH1}
\dot{x}=(\beta_1/2)(x^2-y^2-1) +\beta_2 y(y+c),\,\quad
\dot{y}= \beta_2 y(y+c),
\end{equation*}
with $\beta_1 \beta_2 \neq 0$. This is system (H.1) of Theorem \ref{teo1}.
If $a\neq 0$ and $b=0$ we take $a=1$ and system \eqref{eq1} satisfies $P=k_2 f_2$
and $2y\,Q=2x P-k_1 f_1$, where $k_j=\alpha_j+ \beta_j x+ \gamma_j y$, for $j=1,2$.
Since $Q$ is a polynomial in the parameters of the system it must satisfy one of the
following conditions
\begin{gather*}
s_1=\{c= -1, \, \alpha_1= 0,\,\beta_1= 2 \alpha_2,\,\beta_2=\alpha_2\},\\
s_2=\{c= 1, \, \alpha_1= 0,\,\beta_1= -2 \alpha_2,\,\beta_2= -\alpha_2\},\\
s_3=\{\alpha_1= 0,\,\alpha_2= 0,\,\beta_1= 0,\,\beta_2= 0\}.
\end{gather*}
Applying the change of coordinates $x=-X, y= Y$ we conclude that case $s_1$ and $s_2$
provide equivalent systems. Moreover we can summarize solutions $s_1$ and $s_3$ in the
unique system
\begin{equation}
\label{proofH2}
\dot{x}=(x+c)(\alpha_2 x+ \gamma_2 y+\alpha_2),\quad
\dot{y}= -(\gamma_1/2)(x^2-y^2-1)+x(\gamma_2 x +\alpha_2 y+ c \gamma_2),
\end{equation}
with $\alpha_2(c+1)=0$. System \eqref{proofH2} corresponds to system (H.2) of Theorem \ref{teo1}.
If $a b \neq 0$ we assume $b=1$ and consider three cases, according to the conditions of
Theorem \ref{CTheo}. Note that condition $(i)$ of Theorem \ref{CTheo} holds because
$\nabla f_1(x,y)=(2x,-2y)$ and $\nabla f_2(x,y)=(a,1)$, where $\nabla$ indicates
the gradient. Condition (ii) also holds. However condition (iv) is not verified when $a^2-1=0$.
Indeed in this case $f_1=(x+y)(x-y)-1$ and $f_2=(y\pm x)+c$. Condition $(iii)$ does not
hold when $c^2=a^2-1$ since the straight line $f_2= y+a x+c=0$ is tangent to the hyperbola.
The proof of this last statement can be done analogously as for the systems of type (E).
Hence when $a^2 - 1 =0$ or $c^2 = a^2 -1$ Theorem \ref{CTheo} does not hold and we split
the study of systems of type $(H)$ for $ab \neq0$ in three cases:
$a^2 - 1=0$, $c^2 = a^2 -1$ and $(a^2 - 1)(c^2- a^2+1) \neq0 $.
For the first two cases we apply Propositions \ref{prop10} and \ref{prop1} to conclude
that $f_1$ is an algebraic invariant curve of a quadratic system \eqref{eq1} and it can
be written as
\begin{equation}
\label{proofH}
\dot{x}=\frac A2 (x^2-y^2-1) - 2y(p+q x+r y),\,
\dot{y}= -\frac B2 (x^2-y^2-1)-2 x (p+q x+r y),
\end{equation}
where $A, B,p,q,r \in \mathbb{R}$. Fixing the cofactor of $f_2=0$ as $k_2=\alpha+ \beta x+ \gamma y$,
where $\alpha, \beta, \gamma \in \mathbb{R}$ and using system \eqref{proofH} we
solve \eqref{algebraicinv}. First considering $a=-1$ (the case $a=1$ is analogous except
by a reflection) equation \eqref{algebraicinv} has two possible solutions
\begin{gather*}
s_1=\{B= -A,\,c= 0,p= \alpha/2, \,q=\beta/2, \,r= \gamma/2\},\\
s_2=\{B= -A+2 c\alpha , \,p= (\alpha c -\beta )/2, \,q=(\beta -c\alpha )/2,
r=-(\beta+ c \alpha)/2, \,\gamma = -\beta \}.
\end{gather*}
Using the two above solutions we obtain the system
\begin{align*}
\dot{x}=&(A/2) (x^2-y^2-1)-y (\alpha -c\beta+x (\beta -c\alpha)+y (\gamma -c\alpha)),\\
\dot{y}=&(A/2)(x^2-y^2-1)-x (\alpha -c\beta+\beta x+y (\gamma -c\alpha))+c\alpha (y^2+1),
\end{align*}
with $c (\gamma+\beta)=0$. This is system (H.3) of Theorem \ref{teo1}.
Now considering $c^2=a^2-1$ we investigate the conditions that must be satisfied by
the parameters of system \eqref{proofH} in order that $f_2=y+ ax\pm \sqrt{a^2-1}$
be an invariant curve. Without loss of generality we can assume $c=\sqrt{a^2-1}$.
Equation \eqref{algebraicinv} has one solution, namely
\begin{gather*}
B=a A-2 \alpha\sqrt{d}, \quad p=(\beta\sqrt{d}-a \alpha)/2,\quad r= -\beta/2, \\
q=(\alpha\sqrt{d}-a \beta )/2,\quad \gamma =a \beta-\alpha\sqrt{d},
\end{gather*}
where $d=a^2-1$. Replacing it in \eqref{proofH} we obtain
\begin{align*}
\dot{x}=&\frac A2(x^2-y^2-1)+y(a \alpha-\beta\sqrt{d} +x (a \beta - \alpha \sqrt{d}) +\beta y),\\
\dot{y}=&\frac A2 a(x^2-y^2-1)+x(a \alpha- \beta \sqrt{d}+a \beta x+ \beta y)-\alpha\sqrt{d}(y^2+1),
\end{align*}
where $d=a^2-1$, and this systems corresponds to system (H.4) of Theorem \ref{teo1}.
Finally if $(a^2-1)(c^2-a^2+1)\neq 0$, applying Theorem \ref{CTheo} we obtain the system
\begin{align*}
\dot{x}=&-\alpha_2 (x^2-y^2-1)+2 \alpha_1 y (y+a x+c),\\
\dot{y}=&a\alpha_2 (x^2-y^2-1)+2 \alpha_1 x (y+a x+c),
\end{align*}
which is system (H.5) of Theorem \ref{teo1}.
\subsection{Systems of type (P)} Let $f=(y-x^2)(ax+by+c)=0$ be an invariant cubic of system
\eqref{eq1}. When $b=0$ we can assume $f=x(y-x^2)$. Indeed if $b =0$ we take $a=1$ and
make the change of coordinates $x = X-c, y = Y-2cX+c^2$. Using that $f_2=x=0$ is an
invariant straight line we have $P=k_2 f_2$ with $k_2=\alpha_2+\beta_2 x+\gamma_2 y$,
and a quadratic system \eqref{eq1} can be written as
\begin{equation}
\label{proofP1}
\dot{x}=x (\alpha_2+\beta_2 x+\gamma_2 y),\quad
\dot{y}=\alpha_1( y-x^2)+ 2 \alpha_2 x^2+2y(\beta_2 x+\gamma_2 y).
\end{equation}
If $b \neq 0$ and $a=0$ we can take $b=1$ and proceed as in systems of type (H) and (E),
then we obtain the system
\begin{equation}
\label{proofP2}
\dot{x}=-\beta_1(y-x^2)+y(\beta_2+\gamma_2 x)+(\alpha_2+\gamma_2 c)x+c \beta_2,\quad
\dot{y}=2(y+c) (\alpha_2+\beta_2 x+\gamma_2 y),
\end{equation}
with $c\alpha_2=0$. Observe that when $c=0$ the invariant line is $y=0$ and when
$\alpha_2=0$ it is $y+c=0$.
If $ab \neq0$ and $f_2=y\pm a x + a^2/4$, $f_2=0$ is tangent to the parabola.
In this case we can assume $f_2=y+ax+a^2/4$ (the other case is a reflection).
Applying the change of coordinates $x = -X-a/2$ and $y = Y+a X+a^2/4$ the cubic
$f=(y-x^2)(y+ a x+a^2/4)$ becomes $f=(Y-X^2)Y$, which already has been studied above.
Indeed it corresponds to system \eqref{proofP2} with $c=0$.
Otherwise there is no tangency between the straight line and the parabola, and we apply
Theorem \ref{lema7teo} to get the differential system
\begin{equation}
\label{proofP4}
\dot{x}=-(y-x^2)-\alpha(y+a x+c),\quad
\dot{y}=a (y-x^2)-2 \alpha x (y+a x+c).
\end{equation}
Systems \eqref{proofP1}, \eqref{proofP2} and \eqref{proofP4} correspond to systems (P.1), (P.2)
and (P.3) of Theorem \ref{teo1}, respectively.
\subsection{Systems of type (LV)} In this case $f=x y(a x+b y+c)=0$ is the invariant curve
and except by a rotation we can assume $b=1$. We consider different cases according to
$ac =0$ or $ac\neq0$. Note that if $c = 0$ hypothesis (iii) of Theorem \ref{CTheo} is not
valid, whereas $a = 0$ breaks the hypothesis (iv).
When $c=0$ and $a\neq 0$, doing the change of coordinates
$x = -\frac{Y}{\sqrt[3]{a^2}}$, $y = \sqrt[3]{a} X$ the cubic becomes $F=X Y(Y-X)$.
So using Proposition \ref{prop1} the differential system can be written as
\begin{equation}
\label{LV1}
\dot{x}=x (p_1+q_1 x+r_1 y), \quad
\dot{y}=y(p_2+q_2 x+r_2 y).
\end{equation} If \eqref{LV1} has $f_3=y-x$ as an invariant curve with cofactor
$k=\alpha+\beta x+\gamma y$, then equation \eqref{algebraicinv} must be satisfied.
Solving it we obtain
\[
s_1=\{p_2= \alpha ,\, r_2= \beta -q_2+r_1, \,q_1= \beta ,\,p_1
= \alpha , \,\gamma = \beta -q_2+r_1\}.
\]
Replacing in \eqref{LV1} and writing $q=q_2$, $r=r_1$ we obtain system (LV.1).
Now if $c=a=0$ then $f_2=y=0$ is a double line, and it is not difficult to see that we
can write the system as
\begin{equation}
\label{LV2}
\dot{x}=x (p+q x+r y), \quad \dot{y}=y^2.
\end{equation}
Finally, when $a = 0$ and $c \neq 0$, doing the change of coordinates $x = X/c^2, y = cY-c$
the cubic $f=0$ becomes $F=X Y(Y-1)$. So without loss of generality we can work with
$f_3=y-1$. Again the idea is to write the system as in \eqref{LV1}, and see what are the
conditions in order that $f_3=0$ to be an invariant curve for such system.
Solving equation \eqref{algebraicinv} and replacing the solutions in \eqref{LV1} we obtain
\begin{equation}
\label{LV22}
\dot{x}=x (p+q x+r y), \quad \dot{y}=y(y-1).
\end{equation}
Systems \eqref{LV2} and \eqref{LV22} can be summarized as
\begin{equation*}
\dot{x}=x (p+q x+r y), \quad \dot{y}=y(y+c),
\end{equation*}
with $c=0$ or $c=-1$. This is exactly system (LV.2) of Theorem \ref{teo1}.
In the last case, $a\,c\neq 0$ the invariant cubic is $f=x y\,(y+a x+c)=0$ and by
the geometry to the curves we can assume $a<0$ and $c<0$.
Applying Theorem \ref{CTheo} we obtain the system
\begin{equation*}
\dot{x}=-\alpha_2 x(y+a x+c)-\alpha_3 x y, \quad
\dot{y}=\alpha_1 y(y+a x+c)+a\,\alpha_3 x y.
\end{equation*}
Note that we can take $\alpha_3=1$. Doing $\alpha=\alpha_2,\, \beta=\alpha_1$ we obtain
system (LV.3).
\subsection{Systems of type (RPL)}
Here the invariant cubic is $f=f_1f_2f_3=0$ where $f_1=x+1$, $f_2=x-1$ and $f_3= a x + b y +c$.
When $b=0$ we apply Proposition \ref{prop1} (case (RPL)), then it is easy to see that
the corresponding normal form has one additional invariant curve $f_3=0$ as invariant
straight line if and only if it is a multiple of $f_1$ or $f_2$.
However we cannot consider any of these cases because if the system has $f_2$ as an
invariant double straight line for example, then there would be a change of coordinates
so that the system would be written as\begin{equation*}
\dot{x}= (x-1)(x+1)^2, \quad
\dot{y}= Q(x,y),
\end{equation*}
then having degree $3$ instead of $2$.
When $b \neq0$ we can fix $b=1$. In this case the cubic $f=(x^2-1)(y+ax+c)=0$ can be
reduced to $F=y(x^2-1)$ by change of coordinates $x = X$, $y = Y- a X-c$.
If the quadratic differential system \eqref{eq1} has the invariant curve $f=y(x^2-1)=0$,
then $f_1=0$ and $f_2=0$ are invariant curves and by Proposition \ref{prop1} such system
can be written as
\begin{equation*}
\label{RPL1}
\dot{x}=x^2-1, \quad
\dot{y}=Q(x,y),
\end{equation*}
where $Q(x,y)$ is an arbitrary polynomial of degree $2$. Imposing that $f_3=y = 0$ is
an additional invariant curve with cofactor $k_3 = \alpha+\beta x+\gamma y$, the above
system must satisfy $Q(x,y)=y( \alpha+\beta x+\gamma y)$. This expression justify the normal
form given in (RPL) of Theorem \ref{teo1}.
\subsection{Systems of type (DL)}
These systems have a double straight line as invariant curve which can be taken as $f_1=x$.
We write $f_2=a x+b y+c$ and use the normal form of a system having $f=f_1f_2=0$ as an
invariant cubic. For such normal form, if $b=0$ then $f_2=0$ is an invariant straight
line if and only if $c=0$ but in this case the system cannot have a triple invariant straight
line.
If $b\neq0$ we can take $b=1$ and $f=x^2(y+a x+c)$. Doing the change
$x = X$, $y = Y- a X-c$ the function $f$ can be written as $F=X^2 Y$.
Hence it is enough to consider $f_2=y$. By Proposition \ref{prop1} a quadratic
system \eqref{eq1} can be written as
\begin{equation*}
\dot{x}=x^2, \quad
\dot{y}=Q(x,y),
\end{equation*}
where $Q(x,y)$ is an arbitrary polynomial of degree $2$.
Imposing that $f_2 = 0$ is an additional invariant curve with cofactor
$k_2 = \alpha+\beta x+\gamma y$, we conclude that $Q(x,y)=y(\alpha+\beta x+\gamma y)$.
This expression justify the normal form given by (DL).
\subsection{Systems of type (CPL)} The proof for this case is analogous to the case (DL)
so we will omit some details. In short the cubic is given by $f=f_1f_2f_3 = 0$ where
$f_1=x+i$, $f_2=x-i$ and $f_3=ax+by+c$. In order for $f_3=0$ to be an invariant curve
with $b=0$ it is necessary that $c = \pm i$. So $b \neq 0$ and we assume $b = 1$.
This reduce $f$ to the cubic $F=y(x^2+1)$ and then we obtain the normal form (CPL)
described in Theorem \ref{teo1}.
\subsection{Systems of type (p)}
In this case the cubic is given by $f=(x^2+y^2)(a x+b y+c)=0$ and except by a rotation we
can assume $b=1$. When $c = 0$ the three curves intersect at the same point and the
conditions of Theorem \ref{CTheo} are not satisfied. But if $c = 0$ doing the change of
coordinates
$$
x = -\frac{X}{\sqrt[3]{(a^2+1)^2}}+\frac{a Y}{\sqrt[3]{(a^2+1)^2}}, \quad
y = \frac{a X}{\sqrt[3]{(a^2+1)^2}}+\frac{Y}{\sqrt[3]{(a^2+1)^2}},
$$
the cubic $f=(x^2+y^2)(y+a x)=0$ is reduced to $f=Y(X^2+Y^2)$.
Now using that system \eqref{eq1} has $f_3=y=0$ as a third invariant curve it follows
that $Q(x,y)=k_3 f_3$ where $k_3=\alpha_3+\beta_3 x+\gamma_3 y$ is the cofactor of $f_3$.
Moreover $f_1f_2=0$ is also an invariant curve then we must have
\[
2xP(x,y)+2yQ(x,y)=k(x,y) (x^2+y^2),
\]
with $k(x,y)=\alpha+\beta x+\gamma y$ being the sum of the cofactors of $f_1$ and $f_2$.
So a quadratic system \eqref{eq1} can be written as
\begin{equation*}
\dot{x}= (\beta/2) (x^2+y^2)-\beta_3 y^2+x (\alpha_3+\gamma_3 y), \quad
\dot{y}=y (\alpha_3+\beta_3 x+\gamma_3 y),
\end{equation*}
which is exactly system (p.1) of Theorem \ref{teo1}.
When $c \neq 0$ we apply Theorem \ref{CTheo} and conclude that a quadratic system \eqref{eq1}
can be written as
\begin{equation}
\label{p.2}
\begin{aligned}
& \dot{x}=-\alpha_3 (x^2+y^2)-((\alpha_2+\alpha_1)y-i(\alpha_2-\alpha_1)x) (y+a x+c), \\
& \dot{y}=a \alpha_3 (x^2+y^2)+((\alpha_2+\alpha_1)x-i(\alpha_2-\alpha_1)y) (y+a x+c),
\end{aligned}
\end{equation}
with $\alpha_1, \alpha_2$ and $\alpha_3 \in \mathbb{C}$. Writing $\alpha_j=m_j+i\, n_j$
with $m_j, n_j \in \mathbb{R}$ and using that such system have real parameters we conclude that
$m_2=m_1$, $n_2=-n_1$ and $n_3=0$. Replacing this conditions in \eqref{p.2} we obtain the system
\begin{equation*} %\label{p.21}
\begin{aligned}
\dot{x}=& -m_3 (x^2+y^2)+2( n_1 x-m_1 y) (y+a x+c), \\
\dot{y}=& a m_3 (x^2+y^2)+2(m_1 x+n_1 y) (y+a x+c).
\end{aligned}
\end{equation*}
Note that if $m_3=0$ then the system has a common factor, so we can take $m_3=2$.
After a rescaling of the time and writing $\alpha=m_1,\beta=n_1$ we obtain system (p.2).
It follows from the previous study the proof of Theorem \ref{teo1}.
\section{Proof of Theorems \ref{phaseportraits} and \ref{theo2}} \label{secanalysis}
In this section we investigate planar quadratic systems with algebraic invariant cubics
having Darboux invariant. Moreover we investigate the phase portraits, in
the Poincar\'e disc, of such systems.
\begin{proposition} \label{E}
Each real planar quadratic differential system with a real invariant ellipse and an
invariant straight line having a Darboux invariant can be written, after an affine
change of coordinates, as system {\rm (E.2)} with $c=-1$, $\beta_1=2\beta_2$, $\alpha_2 \neq 0$.
Moreover, such systems have the Darboux invariant of the form
\[
I_2(t,x,y)=e^{-t} (y-1)^{1/\alpha_2} (x^2+y^2-1)^{-\frac{1}{2 \alpha_2}}.
\]
and, they have only two non equivalent phase portraits, see phase portraits
{\rm EL.2.1} and {\rm EL.2.2} of Figure \ref{pp_1}.
\end{proposition}
\begin{proof}
If follows from the reducible cubic classification that we can fix $f_1=x^2+y^2-1=0$
as the real ellipse and by Theorem \ref{teo1} there are only two families of systems having
$f_1=0$ and a straight line as invariant curves (E.1) and (E.2). We shall prove later
that (E.1) does not admit a Darboux invariant. Now we study system (E.2).
By Proposition \ref{prop3} system (E.2) has a Darboux invariant if there exist
$\lambda_1, \lambda_2 \in \mathbb{R}$ not both equal to zero such that \eqref{igualdade de darboux}
holds with $s \in \mathbb{R} \setminus \{0\}$ and $k_1, k_2$ being the cofactors of
$f_1=0$ and $f_2=y+c=0$, respectively. But for system (E.2) we must have $\alpha_2=0$
or $c=-1$. If $\alpha_2=0$ the cofactors are $k_1=\beta_1 x$ and $k_2= \beta_2 x$
and the equation $\lambda_1 k_1+\lambda_2 k_2+s=0$ has no solution for $s \neq 0$.
Hence if $\alpha_2=0$ system (E.2) has no Darboux invariant.
If $\alpha_2 \neq 0$ and $c=-1$ then the cofactors are $k_1=\beta_1 x+2 \alpha_2 y$ and
$k_2=\alpha_2+\beta_2 x+ \alpha_2 y$ and the unique solution of \eqref{igualdade de darboux},
with $s \neq 0$ is
\begin{equation}
\label{s1e1}
\beta_1= 2\beta_2, \quad s= -\alpha_2 \lambda_2, \quad \lambda_1= -\lambda_2/2.
\end{equation}
Taking $\lambda_1=1/\alpha_2$ and replacing \eqref{s1e1} in system (E.2) we obtain system
\begin{equation}
\label{de.1}
\dot{x}=\beta_2(y-1)+x(\beta_2 x+\alpha_2 y),\quad
\dot{y}=(y-1) (\alpha_2+\beta_2 x+\alpha_2 y),
\end{equation}
which has the Darboux invariant
\[
I_2(x,y,t)=e^{-t} (y-1)^{1/\alpha_2} (x^2+y^2-1)^{-\frac{1}{2 \alpha_2}}.
\]
To study the global phase portrait of system (E.2) we start considering its finite
singularities. Note that \eqref{de.1} has at most three finite singularities,
namely $z_1=(0, 1)$, $z_2=(-1/\beta_2,\, 1)$ and
$z_3=\big(-\frac{2\beta_2}{\beta_2^2+1}, \,\frac{\beta_2^2-1}{\beta_2^2+1}\big)$.
The eigenvalues associated to $z_1$ are $2$ and $1$, if $\beta_2 \neq 0$, the
eigenvalues associated to $z_2$ are $-1$ and $1$ and the eigenvalues of $z_3$ are $-1$ and $-2$.
So for $\beta_2 \neq 0$ $z_1$, $z_2$ and $z_3$ are an unstable node, a saddle and a stable
node, respectively. When $\beta_2=0$ we have only $z_1$ and $z_3$ as finite singularities.
In the local chart $U_1$ the compactified system is
\begin{equation} \label{compact1}
\dot{u}=-v (\beta_2+\beta_2 u^2-\beta_2 u v+v),\quad
\dot{v}=-v (\beta_2+\beta_2 u v+u-\beta_2 v^2),
\end{equation}
so $v=0$ is a common factor, this means that $v=0$ is a line of singular points.
Eliminating the common factor $v$, system \eqref{compact1} has no singular points
if $\beta_2 \neq 0$. Otherwise $u_1=(0,0)$ is a singular point with eigenvalues $-1$
and $1$ which implies that the origin is a hyperbolic saddle besides the line of singular points.
In the local chart $U_2$ the compactified system is written as
\begin{equation*}
\dot{u}=v(\beta_2+\beta_2 u^2+u v-\beta_2 v),\quad
\dot{v}= v(v-1)(\beta_2 u+v+1).
\end{equation*}
With a rescaling of the time the common factor $v$ is eliminated and we can see that $(0,0)$
is not a singular point of the compactified system.
Note that if $\beta_2=0$ there are an additional invariant straight line given by $y+1=0$.
From the study of the finite and infinite behavior of system (E.2) we conclude that such
system has two non-equivalent phase portraits when $c=-1$: EL.2.1, if $\beta_2 \neq 0$ and
EL.2.2, if $\beta_2=0$. See Figure \ref{pp_1}.
\end{proof}
\begin{proposition} \label{H}
Each real planar quadratic differential system with an invariant hyperbola and an invariant
straight line having a Darboux invariant can be written, after an affine change of coordinates,
as
\begin{itemize}
\item [(i)] system {\rm (H.2)} with $\alpha_2\neq 0$, $c=-1$ and $\gamma_1=2\gamma_2$.
Its Darboux invariant is
\[
I_3(x,y,t)=e^{-\alpha_2 t} (x^2-y^2-1)^{-1/2}(x-1) .
\]
\item[ii)] system {\rm (H.3)} with $A \alpha \neq 0$, $c=0$ and $\beta=-\gamma$.
Its Darboux invariant is
\[
I_4(x,y,t)=e^{-A \alpha t}(x^2-y^2-1)^{\gamma} (y-x)^{A} .
\]
\item[(iii)] system {\rm (H.3)} with $\alpha \neq 0$ and $\beta=\gamma=0$. Its Darboux invariant is
\[
I_5(x,y,t)=e^{\alpha t}(y-x+c)^{-1}.
\]
\item[(iv)] system {\rm (H.4)} with $\alpha \neq 0$ and $A=2\beta$. Its Darboux invariant is
\[
I_6(x,y,t)=e^{- \alpha t} (x^2-y^2-1)^{-1/2} (y+ a x-\sqrt{a^2-1}).
\]
\end{itemize}
Moreover there are 12 non-equivalent phase portrait in the Poincar\'e disc of these systems.
They are in Figure \ref{pp_1} {\rm HL.2.1--HL.2.3, HL.3.1--HL.3.9.}
\end{proposition}
\begin{proof}
Fixing $f_1=x^2-y^2-1=0$, Proposition \ref{prop3} says that system (H.2) has a Darboux invariant if equation \eqref{igualdade de darboux} holds for $\lambda_1, \lambda_2$ not both zero, where $s \in \mathbb{R} \setminus \{0\}$, and $k_1, k_2$ are cofactors of $f_1=0$ and $f_2=x+c=0$, respectively. Moreover $c=-1$ or $\alpha_2=0$ in system (H.2). For $\alpha_2=0$ we have $k_1=\gamma_1 y$ and $k_2=\gamma_2 y$ and the equation $\lambda_1 k_1+ \lambda_2 k_2+s=0$ has no solution with $s \neq 0$. So in this case system (H.2) has no Darboux invariant. If $\alpha \neq0$ and $c=-1$ then $k_1=2 \alpha_2 x+\gamma_1 y$ and $k_2=\alpha_2+\alpha_2 x+\gamma_2 y$ and \eqref{igualdade de darboux} has a unique solution
$ s= -\alpha_2 \lambda_2,\,\gamma_1= 2 \gamma_2,\,\lambda_1= -\lambda_2/2.$
The proof of (i) follows taking $\lambda_2=1$ and replacing $\gamma_1= 2 \gamma_2$ in system (H.2), so we obtain that system
\begin{equation}
\label{dh.1}
\dot{x}=(x-1) (\alpha_2+\alpha_2 x+\gamma_2 y),\quad
\dot{y}=-\gamma_2 (x^2-y^2-1)+x (-\gamma_2+\gamma_2 x+\alpha_2 y),
\end{equation}has the Darboux invariant
\[
I_3(x,y,t)=e^{-\alpha_2 t} (x^2-y^2-1)^{-1/2}(x-1) . \]
To prove (ii) and (iii) we study system (H.3) considering two cases:
$c=0$ and $\beta=-\gamma$. It is easy to see that if $c=0$ (H.3) has a Darboux invariant
when $\alpha \neq 0$ and $\beta=-\gamma$. In this case we obtain
\begin{equation*}% \label{dh.2}
\dot{x}=(A/2)(x^2-y^2-1)-y (\alpha -\gamma x+\gamma y),\quad
\dot{y}=(A/2)(x^2-y^2-1)-x (\alpha -\gamma x+\gamma y),
\end{equation*}
with the Darboux invariant
\[
I_4(x,y,t)=e^{-A \alpha t}(x^2-y^2-1)^{\gamma} (y-x)^{A}.
\]
If $\beta = -\gamma$, system (H.3) has a Darboux invariant only when $\gamma= 0$ and
$\alpha \neq 0$. In this case, we obtain to system
\begin{align*}% \label{dh.3}
\dot{x}&=(A/2)(x^2-y^2-1)-\alpha y (1 -c x-c y),\\
\dot{y}&=(A/2)(x^2-y^2-1)-\alpha x (1 -c y) +c\,\alpha (y^2+1),
\end{align*}
with the Darboux invariant
\[
I_5(x,y,t)=e^{\alpha t}(y-x+c)^{-1}.
\]
The study of (iv) follows from system (H.4) where the invariant line is
$f_2=y+a x-\sqrt{a^2-1}=0$. In this case the unique solution of \eqref{igualdade de darboux} is
$s= -\alpha \lambda_2$, $A = 2\beta$, $\lambda_1= -\lambda_2/2$.
So taking $\lambda_2=1$ we obtain the Darboux invariant
\[
I_6(x,y,t)=e^{-\alpha t} (x^2-y^2-1)^{-1/2} (y+a x-\sqrt{a^2-1}).
\]
Now, let us study of possible phase portraits of system \eqref{dh.1}.
Since $\alpha_2 \neq 0$ we can take $\alpha_2=1$ and the transformation $x = X, y = -Y$
takes the system with parameter $\gamma_2$ to the system with parameter $-\gamma_2$.
So we may also assume $\gamma_2 \geq0$.
Considering the finite singularities, if $\gamma_2 \not\in \{0, 1\}$ system \eqref{dh.1}
has three finite singularities, namely $z_1=(0, 1)$, $z_2=(1, -1/\gamma_2)$ and
$z_3=\big(\frac{\gamma_2^2+1}{\gamma_2^2-1}, -\frac{2\gamma_2}{\gamma_2^2-1} \big)$.
The eigenvalues associated to $z_1$ are $2$ and $1$, if $\beta_2 \neq 0$, the
eigenvalues associated to $z_2$ are $-1$ and $1$ and the eigenvalues of $z_3$ are
$-1$ and $-2$. So for $\gamma_2 \not\in \{0, 1\}$ $z_1$, $z_2$ and $z_3$
are respectively, an unstable node, a saddle and a stable node. When $\beta_2=0$ we
have only $z_1$ and $z_3$ as finite singularities.
In the local chart $U_1$ the compactified system is
\begin{equation} \label{compact2}
\dot{u}=v (-\gamma_2+\gamma_2 u^2+u v+\gamma_2 v),\quad
\dot{v}=v(v-1) (\gamma_2 u+v+1),
\end{equation}
so $v$ is a common factor, this means that $v=0$ is a line of singular points.
Eliminating the common factor $v$, system \eqref{compact2} has no singular points
if $\gamma_2 \neq 1$. Otherwise $u_1=(-1,0)$ is a singular point with eigenvalues $-2$
and $-1$, which implies that $u_1$ is a hyperbolic stable node. Moreover if $\gamma_2=0$
there an additional invariant straight line given by $x+1=0$.
In the local chart $U_2$ the compactified system is written as
\begin{equation*}
\dot{u}=-v (\gamma_2-\gamma_2 u^2+\gamma_2 u v+v),\quad
\dot{v}=-v (\gamma_2+\gamma_2 v^2-\gamma_2 u v+u).
\end{equation*}
So applying a rescaling of the time to eliminate the common factor $v$, we obtain that
the origin is a singular point of the compactified system if and only if $\gamma_2=0$.
In this case $(0,0)$ is a hyperbolic saddle.
It is easy to see that if $\gamma_2 \in (0,1)$ the singularities $z_1$ and $z_3$ are in
distinct branches of the hyperbola, and if $\gamma_2 \in (1, +\infty)$ they are in the same
branch as shows Figure \ref{hipNewm1}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.6]{fig8} % equivalence_hip_1.eps}
\caption{Possible phase portraits of sytem \eqref{dh.1} when $\gamma_2 \not\in \{0,1\}$.}
\label{hipNewm1}
\end{figure}
From \cite[Theorem 1.43]{Llibre} (Markus-Neumann-Peixoto Theorem) we conclude that
these two phase portraits are topologically equivalent. By continuity and the study done
previously we conclude that system of type (H.2) having a Darboux invariant can have
three non-equivalent phase portrait. The case $\gamma_2 \neq 0,1$ corresponds to HL.2.1
in Figure \ref{pp_1} and when $\gamma_2=1$ or $\gamma_2=0$ we have the phase portraits HL.2.2
and HL.2.3 of Figure \ref{pp_1}, respectivelly.
\end{proof}
Now we study the global phase portrait of system (H.3). Remember that the parameters of (H.3)
must satisfy $c(\gamma+\beta)=0$. We start considering $c=0$, then the differential system is
\begin{equation} \label{h.3}
\begin{gathered}
\dot{x}=(A/2)(x^2-y^2-1)-y (\alpha -\gamma x+\gamma y),\\
\dot{y}=(A/2)(x^2-y^2-1)-x (\alpha -\gamma x+\gamma y),
\end{gathered}
\end{equation}
which has $f_1=x^2-y^2-1=0$ and $f_2=y-x=0$ as invariant algebraic curves.
Since $\alpha \neq 0$ we can take $\alpha=1$ and the transformation $x = -X, y = -Y$
allows to assume $A>0$.
If $\gamma \neq 0$ then $z_1=(-A/2,-A/2)$ and
$z_2=\left(({\gamma ^2+1})/({2 \gamma}),({\gamma^2-1})/({2 \gamma})\right)$
are the two finite singular points. If $\gamma=0$ exists only one finite singular point.
The eigenvalues associated to $z_1$ are $-1$ and $1$ so $z_1$ is a saddle.
The eigenvalues associated to $z_2$ are $A/\gamma$ and $-1$, so $z_2$ is a stable node
if $\gamma<0$, and a saddle if $\gamma>0$. Moreover $z_1$ is on the straight line and $z_2$
is on the hyperbola.
In the local chart $U_2$, we have system
\begin{align*}
& \dot{u}=(1/2) (u-1)(A v^2-(A+2\gamma) u^2+2 u v+2 v+A+2 \gamma),\\
&\dot{v}=(1/2)v(A v^2-(A+2\gamma) u^2+2 \gamma u+2 u v+A),
\end{align*}
and the origin is a singular point only when $A+2 \gamma=0$ but in this case the line $v=0$
is filled up of singular points.
In the local chart $U_1$, we have system
\begin{align*}
& \dot{u}=(1/2) (u-1)((A+2 \gamma) u^2+A v^2 +2 u v+2 v-A-2 \gamma),\\
&\dot{v}=(1/2)v((A +2 \gamma )u^2+A v^2+2 u v-2 \gamma u-A),
\end{align*}
which has the infinity filled up by singularities when $A+2 \gamma=0$, otherwise,
there are two singularities $u_1=(-1,0)$ and $u_2=(1,0)$.
Assuming $A+2 \gamma\neq 0$. The point $u_1$ has eigenvalues $2\gamma$ and $2(A+2\gamma)$,
and $u_2$ is linearly zero because the Jacobian matrix of the linear part of the system
evaluated in $u_2$ is null. To decide the local behavior of $u_2$ we must have a \emph{blow up}.
From now on we fix $l_1= \gamma$, $l_2=A+2\gamma$.
After translating the singular point $u_2$ to the origin, making the change of coordinates
$u = U$, $v = U W$ and rescaling the common factor $U$, we obtain
\begin{equation*}
\dot{U}=(1/2) U (A U W^2+(A+2 \gamma )U+2 U W+4 W+2 A+4 \gamma),\quad
\dot{W}=- W (W+\gamma).
\end{equation*}
Note that such system has two singularities when $l_1 l_2 \neq 0$, namely,
$\overline{U_1}=(0,0)$ and $\overline{U_2}=(0,-\gamma)$; one singular point when $l_1=0$
and $l_2 \neq 0$, namely $\overline{U_1}=\overline{U_2}$.
The eigenvalues of $\overline{U_1}$ are $-\gamma$ and $A+2\gamma$, whereas the eigenvalues
of $\overline{U_2}$ are $A$ and $\gamma$. From the combination of the signs of $l_1$ and
$l_2$, as described in Figure \ref{esquema1}, we obtain the possible local behavior
of $\overline{U_1}$ and $\overline{U_2}$.
\begin{figure}[htbp]
\includegraphics[scale=0.9]{fig9} % esquema_h5.eps
\caption{The possible combination of signs of $l_1$ and $l_2$ describe the cases to be
considered for system (H.3) when $c=0$.}
\label{esquema1}
\end{figure}
After blowing down we obtain all possible phase portraits for system (H.3)
when $c=0$. Note that each one is realizable. Indeed, the phase portrait HL.3.2 corresponds
to subcase (1.1) which is realizable with $A=4$ and $\gamma=-1$; HL.3.3 corresponds to subcase
(1.2) which is realizable with $A=1$ and $\gamma=-1$. Notice that if $\gamma \neq 0$ there
is a third invariant straight line, given by $f_3=\gamma (x-y)-1=0$ so HL.3.3 is the only
possible phase portrait for subcase $(1.2)$. The phase portraits HL.3.4 and HL.3.5 correspond,
respectively, to subcases $(2.1)$ and $(3.1)$. The phase portrait HL.3.4 is realizable with
$A=1$ and $\gamma=1$, and HL.3.5 is realizable with $A=1$ and $\gamma=0$.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.7]{fig10} % bd_h_5.eps
\caption{On the left the local phase portrait after blow up. Here they are indexed
according to the signs of $l_1$ and $l_2$. On the right the local behavior at origin after
the \emph{Blow down} for system (H.3).}
\label{bd_h_5}
\end{figure}
It remains to consider $l_2=0$. With this condition the infinity is filled up of singular
points. After eliminating the common factor $v$ we have only one singular point at the local
chart $U_1$. The eigenvalues associated to this point are $2$ and $1$,
so this is an unstable node. By continuity the only possible phase portrait in this case is
HL.3.1 of Figure \ref{pp_1}, which is realizable with $A=2$ and $\gamma=-1$.
Now considering system (H.3) with $\beta+\gamma=0$ we have seen above that the system has
a Darboux invariant when $\beta=\gamma=0$ and $\alpha \neq 0$. Under these conditions the
differential system is
\begin{equation}
\label{systemh6}
\begin{array}{cl}
& \dot{x}=(A/2)(x^2-y^2-1)-\alpha y (1 -c x-c y),\\
&\dot{y}=(A/2)(x^2-y^2-1)+c \alpha(y^2+1)-\alpha x(1-cy).
\end{array}
\end{equation}
Such system has $f_1=x^2-y^2-1=0$ and $f_2=y-x+c=0$ as algebraic invariant curves.
If $c=0$ then we obtain system \eqref{h.3} when $\gamma=0$, so we can take $c \neq 0$ here.
Moreover, doing the transformation $x = -X, y= -Y$ in the algebraic cubic we can assume $c>0$.
Finally, since $\alpha$ is different from zero we can take $\alpha=1$ in \eqref{systemh6}.
System \eqref{systemh6} has two finite singular points, namely $z_1=((2c-A)/2, \, -A/2)$ and
$z_2=((c^2+1)/(2 c), (1-c^2)/(2 c))$. Defining $l_1=c^2-Ac-1$, $l_2=A-c$ and $l_3=A-2c$, we have
$z_1$ coalesces with $z_2$ if and only if $l_1=0$. Moreover the eigenvalues associated to
$z_1$ are $l_1$ and $1$, and the eigenvalues associated to $z_2$ are $-l_1$ and $1$.
So we conclude that $z_1$ is a unstable node and $z_2$ is a saddle if $l_1>0$; $z_1$ is a
saddle and $z_2$, an unstable node, if $l_1<0$ and, if $l_1=0$, $z_1=z_2$ is a saddle-node.
In the local chart $U_1$, system \eqref{systemh6} becomes
\begin{align*}
& \dot{u}=\frac 12 ((A-2c) u^3-A u^2+A u v^2- u(A-2c)-v^2(A-2c)+2 u^2 v-2 v+A),\\
&\dot{v}=\frac 12 v((A-2c) u^2+A v^2-2 c u+2 u v-A),
\end{align*}
which has three singularities $u_1=(-1,0)$ and $u_2=(1,0)$ and $u_3=(\frac{A}{A-2 c},0)$,
if $A\neq 2c$. Note that when $l_3=0$ the point $u_3$ does not exist and $u_1=u_3$ when $l_2=0$.
The eigenvalues associated to $u_1$ are $2l_2$ and $0$, the point $u_2$ has both eigenvalues
equal to $-2c$, and $u_3$ has eigenvalues $0$ and $2cl_2/l_3$. It is not difficult to see that
when $l_2 \neq 0$, $u_1$ and $u_3$ are saddle--nodes. In the local chart $U_2$ the origin
$(0,0)$ is a singular point if and only if $l_3=0$.
Assuming $l_1 l_2 \neq 0$ and considering all possible combinations of the sign of $l_1, l_2$
and $l_3$ we observe that there are some impossible combinations, for instance when $l_2 <0$
we have $l_3<0$. In Figure \ref{esquema_h6} we describe the possible combinations and introduce
a label for each one.
\begin{figure}[httb]
\includegraphics[scale=0.9]{fig11} % esquema_h6.eps}
\caption{The possible combinations of signs of $l_1, l_2$ and $l_3$ for system (H.3)
when $c \neq 0$.}
\label{esquema_h6}
\end{figure}
The case (2.2.1) presents a unique phase portrait, HL.3.6 of Figure \ref{pp_1} and it
is realizable with $A=1/2$ and $c=1$.
In case (2.1.1) we have three possibilities for the finite saddle separatrix $\omega$-limit set:
we can have a connection of separatrix as in HL.3.7; the separatrix can go to the stable node,
generating a phase portrait equivalent to HL.3.6, or the separatrix can go to the parabolic
sector of the saddle node $u_3$ which corresponds to HL.3.8. Moreover HL.3.8 is realizable
with $A=2$ and $c=1/2$, and as we see above, HL.3.6 is realizable with $A=1/2$ and $c=1$.
Since HL.3.6 and HL.3.8 are realizable then by continuity of the parameters we conclude
that HL.3.7 is also realizable.
The analysis of case (2.1.2) can be done as the case (2.1.1) and it has the phase portraits
equivalent to them.
The possible phase portraits of (2.1.3) are also equivalent to the phase portraits of (2.1.1).
Also the case (1.1.1) has a phase portrait equivalent to (2.2.1).
When $l_2=0$ it follows that $l_1, l_3<0$ and in the local chart $U_1$ the singular point
$u_1=u_3$ is non-elementary. After translate this singular point to the origin, making
the change of coordinates $u = U$, $v = U W$ and rescaling the common factor $U$ we obtain
\begin{equation*}
\dot{U}=(U/2) (A U W^2-A U+2 U W-4 W+2 A),\quad
\dot{W}= W (W-A).
\end{equation*}
This system has two singularities $\overline{U_1}=(0,0)$ and $\overline{U_2}=(0,A)$ being
both saddles. Figure \ref{bd_h_6} shows the \emph{blow down}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.7]{fig12} % bd_h_6.eps
\caption{The local phase portrait of system \eqref{systemh6} in the local chart $U_1$
when $l_2=0$. On the left the local phase portrait after blow up. On the right the local
behavior at the origin after blow down.}
\label{bd_h_6}
\end{figure}
To obtain the phase portrait for system \eqref{systemh6} with $l_2=0$ we note that there
is more two invariant straight lines, given by $f_3=x+y=0$ and $f_4=A x+A y-1$.
The finite saddle $z_1$ is on $f_3=0$ and the finite node is on the intersection of $f_2=0$
and $f_4=0$ so by continuity there is only one phase portrait, which is topologically
equivalent to HL.3.3.
Finally it remains to study the case $l_1=0$. Here $l_2<0$ and $l_3<0$ so the only possibility
is the phase portrait HL.3.9 of Figure \ref{pp_1}, which is realizable with $A=0$ and $c=1$.
To conclude the proof of Proposition \ref{H} it remains to study the global phase portrait
of system (H.4) when $A=2\beta$ and $\alpha \neq 0$. In this case we assume $\alpha=1$,
so (H.4) is written as
\begin{align*}
& \dot{x}=\beta x^2+(a \beta-\sqrt{a^2-1}) x y+(a-\sqrt{a^2-1} \beta ) y-\beta ,\\
&\dot{y}=(a \beta-\sqrt{a^2-1})y^2+\beta x y+(a-\sqrt{a^2-1} \beta ) x+(a \beta-\sqrt{a^2-1}).
\end{align*}
Denoting $\delta=a \beta-\sqrt{a^2-1}$ and $\eta=a-\sqrt{a^2-1} \beta $ there are at most
three finite singularities $z_1=(-\delta/\eta, \beta/\eta)$,
$z_2=((\delta \eta -\beta)/(\beta ^2-\delta ^2),(\delta \eta -\beta)/(\beta ^2-\delta ^2))$
and $z_3=((\beta +\delta \eta)/( \beta ^2-\delta ^2),
-(\beta +\delta \eta)/( \beta ^2-\delta ^2))$.
We observe that such singular points never coalesce but if $\eta=0$, $z_1$ does not exist
and if $\beta^2-\delta^2=0$ the same happens with $z_2$ or $z_3$. With respect to the
localization of these points, $z_3$ is the intersection of the hyperbola and the straight line,
$z_1$ is on the straight line and $z_2$ is on the hyperbola.
Moreover it is not difficult to check that $z_1, z_2$ and $z_3$ are hyperbolic points,
being $z_1$ a saddle, $z_2$ a stable node and $z_3$ an unstable node.
Concerning to the behavior at infinity, in the local chart $U_1$ the compactified system
is given by
\begin{equation*}
\dot{u}=v(\eta -\eta u^2+\beta u v+\delta v),\quad
\dot{v}=-v(\beta-\beta v^2 +\eta u v+\delta u),
\end{equation*}
so $v$ is a common factor what means that $v=0$ is a line of singular points.
Eliminating this common line it remains singularities if and only if $\eta=0$ or
$\beta^2-\delta^2=0$. When $\eta=0$ the point $u_1=(-a,0)$ is a saddle. When $\delta=\beta$
the point $u_2=(-1,0)$ is a node with eigenvalues $\eta$ and $2\eta$.
Finally if $\delta=-\beta$ then the point $u_3=(1,0)$ has eigenvalues $-\eta$ and $-2\eta$
so it is a node.
In the local chart $U_2$, the system becomes
\begin{equation*}
\dot{u}=-v (-\eta +\beta v+\eta u^2+\delta u v),\quad
\dot{v}=v(\delta +\beta u+\eta u v+\delta v^2).
\end{equation*}
So eliminating the common factor $v$ the origin is not a singular point.
By the previous study and continuity of the solutions we conclude that there exist three
possible phase portraits and they are topologically equivalent to the ones obtained from
system (H.2) and described in Figure \ref{pp_1}. Indeed when $\eta, \beta^2-\delta^2 \neq 0$
we have the phase portrait HL.2.1, when $\beta^2-\delta^2 = 0$ we have HL.2.2, and the case
$\eta=0$ corresponds to phase portrait HL.2.3.
Before to study the systems of type (P), we present two lemmas that will help to show the
realization or not of the phase portraits that follow.
\begin{lemma} \label{lemma1}
On any straight line which is not composed of orbits the total number of contact points
is at most two for any quadratic system. If there are two such points $p_1$ and $p_2$,
then the orbits intersecting the segment $\infty p_1$ cross in the same sense as the
orbits intersecting $p_2 \infty $, and the opposite sense to the path intersecting $p_1p_2$.
\end{lemma}
\begin{lemma} \label{lemma2}
The straight line connecting one finite singular point and a pair of
infinite singular points in a quadratic system is either formed by
trajectories or a line with exactly one contact point. If this
contact point is the finite singular point, the flow goes in
different directions on each half straight line.
\end{lemma}
The proof of Lemma \ref{lemma1} is in \cite{Copel}.
Lemma \ref{lemma2}, in the case that the pair of infinite singular points are
saddles is in \cite{Sotomayor}. When such a pair are saddle-nodes, the proof
appeared in \cite{artestese}.
\begin{proposition} \label{P}
Each real planar quadratic differential system with an invariant parabola and an invariant
straight line having a Darboux invariant can be written, after an affine change of coordinates,
as
\begin{itemize}
\item [(i)] {\rm (P.1)} with $\alpha_1-2\alpha_2 \neq 0$ and Darboux invariant
\begin{equation*}
I_7(x,y,t)=e^{ (\alpha_1-2\alpha_2)t}(y-x^2)^{-1}x^2 .
\end{equation*}
\item [(ii)] {\rm (P.2)} with $\alpha_2 (\beta_1-\beta_2) \neq 0$, $\gamma_2=c=0$ and
Darboux invariant
\begin{equation*}
I_8(x,y,t)= e^{2 \alpha_2 (\beta_1-\beta_2)t}(y-x^2)^{\beta_2}y^{-\beta_1} ,
\end{equation*}
\item [(iii)] {\rm (P.2)} with $c\,\gamma_2 \neq 0$, $\beta_1=\beta_2$, $\alpha_2=0$
and Darboux invariant
\begin{equation*}
I_9(x,y,t)= e^{-2 c \gamma_2t}(y-x^2) (y+c)^{-1},
\end{equation*}
Moreover there are 41 non-equivalent phase portrait in the Poincar\'e disc for these systems.
They are in Figures \ref{pp_2} and \ref{pp_3}.
\end{itemize}
\end{proposition}
\begin{proof}
We fix the invariant parabola as $f_1=y-x^2=0$. Here we describe in details the proof of the
existence of a Darboux invariant for system (P.2), the other cases are analogous.
System (P.2) is given by
\begin{equation*}
\dot{x}=-\beta_1(y-x^2)+y(\beta_2+\gamma_2 x)+(\alpha_2+\gamma_2 c)x+c \beta_2,\quad
\dot{y}=2(y+c) (\alpha_2+\beta_2 x+\gamma_2 y),
\end{equation*}
where $c\, \alpha_2=0$. If $c=0$ then the additional invariant line is written as $f_2=y=0$
and if $\alpha_2=0$, such line is $f_2=y+c=0$.
System (P.2) has a Darboux invariant if there exist $\lambda_1, \lambda_2$ not all zero
satisfying equation \eqref{igualdade de darboux} with $s \in \mathbb{R} \setminus \{0\}$, and
$k_1, k_2$ being the cofactors of $f_1=0$ and $f_2=0$, respectively.
For $c=0$, $k_1=2 (\alpha_2+\beta_1 x+\gamma_2 y)$ and $k_2=2 (\alpha_2+\beta_2 x+\gamma_2 y)$.
Equation \eqref{igualdade de darboux}, with $s \neq 0$ has the solution
\begin{equation*}
s= -2 \alpha_2 (\lambda_1+\lambda_2),\,\beta_2= -\beta_1 \lambda_1/\lambda_2,\,\gamma_2= 0,
\end{equation*} Taking $\lambda_1= \beta_2$ and $\lambda_2=-\beta_1$ the solution can be
rewritten as
\begin{equation*}
s= -2 \alpha_2 (\beta_2-\beta_1),\quad \lambda_1 = \beta_2,\quad
\lambda_2 =-\beta_1, \quad \gamma_2= 0,
\end{equation*}
and the Darboux invariant is
\[
I_8(x,y,t)= e^{2 \alpha_2 (\beta_1-\beta_2)t}(y-x^2)^{\beta_2}y^{-\beta_1}.
\]
In this case we assume $\beta_2-\beta_1\neq 0$ otherwise system (P.2) has a common factor.
Moreover if $\alpha_2=c=0$ (P.2) does not admit a Darboux invariant.
When $\alpha_2=0$ then $f_2=y+c$ and the cofactors of $f_1=0$ and $f_2=0$ are, respectively,
$k_1=2 (c \gamma_2+\beta_1 x+\gamma_2 y)$ and $k_2=2 (\beta_2 x+\gamma_2 y)$.
In this case equation \eqref{igualdade de darboux} has only one solution
\[
s= -2 c \gamma_2 \lambda_1,\,\beta_2= \beta_1,\, \lambda_2= -\lambda_1.
\]
So taking $\lambda_1=1$ we obtain the Darboux invariant
\[
I_9(x,y,t)=e^{-2 c \gamma_2t}(y-x^2) (y+c)^{-1}.
\]
From now on we study the possible global phase portraits for systems (P) when they have
a Darboux invariant. We start studying system (P.1). Remember that such system is given by
\begin{equation*}
\dot{x}=x (\alpha_2+\beta_2 x+\gamma_2 y),\quad
\dot{y}=\alpha_1( y-x^2)+ 2 \alpha_2 x^2+2y(\beta_2 x+\gamma_2 y).
\end{equation*}
We consider two cases: $\gamma_2 \neq 0$ and $\gamma_2 =0$. If $\gamma_2 \neq 0$
we assume $\gamma_2=1$. In this last case system (P.1) have at most four singular points,
given by
\begin{gather*}
z_1=(0,0), \quad z_2=(0,-\alpha_1/2),\\
z_3=\Big(-(\beta_2+\sqrt{\beta_2^2-4 \alpha_2})/2,\,
(\beta_2^2-2 \alpha_2 + \beta_2\sqrt{\beta_2^2-4 \alpha_2})/2 \Big),\\
z_4=\Big(-(\beta_2-\sqrt{\beta_2^2-4 \alpha_2})/2,\,
(\beta_2^2-2 \alpha_2- \beta_2\sqrt{\beta_2^2-4 \alpha_2})/2 \Big).
\end{gather*}
Observe that applying the change of coordinates $x = -X, y = Y$ we can assume $\beta_2 \geq 0$.
Let $l_1=\alpha_1$, $l_2=\alpha_2$, $l_3=\beta_2^2-4 \alpha_2-\beta_2 \sqrt{\beta_2^2-4 \alpha_2}$
and $l_4=\alpha_1-2\alpha_2$ be. It follows from Proposition \ref{P} (i) $l_4 \ne0$. Moreover
\begin{itemize}
\item $z_1$ has eigenvalues $l_1$ and $l_2$;
\item $z_2$ has eigenvalues $-l_1$ and $-l_4$;
\item $z_3$ has eigenvalues $l_4$ and
$(\beta_2^2-4\alpha_2+\beta_2 \sqrt{\beta_2^2-4\alpha_2})/2$;
\item $z_4$ has eigenvalues $l_3$ and $l_4$,
\end{itemize}
so $l_1^2 + l_2^2\neq 0$ and the topological type of the finite singular points can be
studied using the Hartman-Grobman Theorem and \cite[Theorem 2.19]{Llibre}.
With respect to the position of the finite singularities, $z_1$ is on the intersection of the parabola and the straight line, $z_2$ is on the straight line, and $z_3, z_4$ are on the parabola.
In the local chart $U_1$, system (P.1) is written as
\begin{equation*}
\dot{u}=u^2+\beta_2 u+(\alpha_1-\alpha_2) u v+2 \alpha_2-\alpha_1,\quad
\dot{v}=-v (\alpha_2 v+u+\beta_2),
\end{equation*}
which has at most two singular points when $v=0$, namely
\begin{equation*}
u_1=(-\beta_2-\sqrt{\beta_2^2+4( \alpha_1-2 \alpha_2)}/2, \,0),\quad
u_2=(-\beta_2+\sqrt{\beta_2^2+4( \alpha_1-2 \alpha_2)}/2, \,0).
\end{equation*}
The eigenvalues associated to $u_1$ are $-\sqrt{\beta_2^2+4l_4}$ and
$-(\beta_2-\sqrt{\beta_2^2+4l_4})/2$ while the eigenvalues associated to $u_2$ are
$\sqrt{\beta_2^2+4l_4}$ and $-(\beta_2+\sqrt{\beta_2^2+4l_4})/2$.
Since we are assuming $\beta_2 \geq 0$ it follows that when $\beta_2^2+4l_4>0$ the
point $u_2$ is a saddle and it is not difficult to see that if $l_4>0$, then $u_1$ is a saddle,
and if $l_4<0$, $u_1$ is a stable node. When $\beta_2^2+4l_4=0$ $u_1$ and $u_2$ coalesce and
we conclude that this point is a saddle-node, using \cite[Theorem 2.19]{Llibre}.
When $\beta_2^2+4l_4<0$ there is no infinite points in the local chart $U_1$.
In the local chart $U_2$ the origin $(0,0)$ is a stable node.
Observe that $l_1, l_2, l_3, l_4, \beta_2^2-4 \alpha_2$ and $\beta_2^2 + 4 l_4$ are
bifurcation surfaces, i.e. where topological changes in the global phase portrait of (P.1)
can happen. To draw all non-equivalent phase portraits of system (P.1) we split the study
in three cases: $\beta_2^2-4 \alpha_2>0$, $\beta_2^2-4 \alpha_2=0$ and $\beta_2^2-4 \alpha_2<0$.
Choosing a representative of each region defined by such surfaces we have a configuration
of finite and infinite points. Considering the behavior of the separatrices of these systems
we obtain all possible phase portraits when $\beta_2^2-4 \alpha_2>0$,
thus we obtain the $40$ phase portraits described in Figures \ref{pppp_1} and \ref{pppp_2}
and the phase portraits $41-50$ of Figure \ref{pppp_3}. We study all these cases bellow.
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig13} % pppp_1.eps
\caption{Phase portraits of system (P.1) when $\gamma_2=1$ and $\beta_2^2-4 \alpha_2>0$.}
\label{pppp_1}
\end{figure}
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig14} % pppp_2.eps}
\caption{Phase portraits of system (P.1) when $\gamma_2=1$ and $\beta_2^2-4 \alpha_2>0$.}
\label{pppp_2}
\end{figure}
Among the phase portraits $1-18$ of Figure \ref{pppp_1}, we claim that $1$ and $3$, as well
as $7$ to $18$, are not realizable. Indeed these $18$ phase portraits, $1-3$ present the possible
combinations when the singular points in the local chart $U_1$ are both saddles.
In the finite part we have $z_1$ and $z_3$ unstable nodes, $z_2$ is a stable node and $z_4$
is a saddle. So we have $l_1, l_2, l_4>0$ and $l_3<0$. In phase portrait $1$ of
Figure \ref{pppp_1}, consider the straight line joining the finite singular point $z_3$
to the infinity singular point $u_1$ as shows Figure \ref{figlemma2}.
We can see that near the singular point $z_3$ but on opposite sides, the vector field has
the same direction, which contradicts Lemma \ref{lemma2}. So the phase portrait $1$
of Figure \ref{pppp_1} is not realizable. With the same argument the portrait $3$
of Figure \ref{pppp_1} is also not realizable. So phase portrait $2$ of Figure \ref{pppp_1}
is the only realizable and corresponds to phase portrait PL.1.1 of Figure \ref{pp_2}.
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig15} % figlemma2.eps}
\caption{The straight line joining the finite singular point $z_3$ to the infinity
singular point $u_1$ in phase portrait 1 of Figure \ref{pppp_1}.}
\label{figlemma2}
\end{figure}
Considering the phase portraits 4--18 of Figure \ref{pppp_1} we shall prove that $7-18$ are
not realizable. First consider the phase portrait $7$ and the straight line joining the middle
point between the infinity singular points $u_1$ and $u_2$ and the middle point between the
finite singular points $z_3$ and $z_4$ as shows Figure \ref{figlemma1}.
By Lemma \ref{lemma1} this line should have at most two points of contact with the vector field,
which does not occur. In Figure \ref{figlemma1} we can see at least four contact points,
represented by the smaller points that are not singularities of the system.
This fact guarantees that the $\omega-$limit set of $u_2$ is the finite point $z_4$ on the
parabola. So phase portraits $7-18$ are not realizable using similar arguments.
So among the phase portraits $4-18$ only $4, 5$ and $6$ are realizable, which correspond,
respectively to phase portraits PL.1.2, PL.1.3 and PL.1.4 of Figure \ref{pp_2}.
The values of the parameters that realize these systems can be found in Table \ref{tabela}.
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig16} % figlemma1.eps}
\caption{The straight line joining the middle point between the infinity singular points $u_1$ and $u_2$ and the middle point between the finite singular points $z_3$ and $z_4$ in phase portrait 7 of Figure \ref{pppp_1}.}
\label{figlemma1}
\end{figure}
The phase portraits $19-20$ in Figure \ref{pppp_1} and 21-26 in Figure \ref{pppp_2} are
topologically equivalent to one of the phase portraits 1--18 in Figure \ref{pppp_1} so they
can be realizable or not, depends on their configuration.
In Table \ref{tabelarelation} we present the relation among the equivalent phase portraits
of system (P.1) when $c \neq 0$. In the case where they are topologically equivalent to
a realizable phase portrait, we need not consider the study again. However if they are
topologically equivalent to a phase portrait which was not realizable, we need to study it.
Considering the same straight line used to prove the non-realization of phase portraits 7--18
of Figure \ref{pppp_1} we apply Lemma \ref{lemma1} to conclude that 21, 22, 25 and 26
of Figure \ref{pppp_2} are not realizable.
The phase portraits 27--31 in Figure \ref{pppp_2} present all the possibilities when
there are four finite singular points and one infinite singular point on the local
chart $U_1$. Phase portraits 27--29 are realizable and correspond to phase portraits
PL.1.5, PL.1.6 and PL.1.7 of Figure \ref{pp_2}. The values of the parameters that
realize these systems can be found in Table \ref{tabela}. Moreover 30 and 31
are topologically equivalent to one of these three phase portraits.
\begin{table}[http]
\caption{Table of relations among all the possible phase portraits of system (P.1)
when $c \neq 0$.}
\label{tabelarelation}
\centering
\begin{tabular}{|c|c|}
\hline
Phase portrait & Topologicaly equiv. \\
$19$ & $2$\\
$20$ & $6$\\
$21$ & $12$\\
$22$ & $9$\\
$23$ & $2$\\
$24$ & $6$\\
$25$ & $12$\\
$26$ & $9$\\
$30$ & $29$\\
$31$ & $29$\\
$35$ & $34$\\
$36$ & $34$\\
$60$ & $50$\\
\hline
\end{tabular}
\end{table}
Finally if there are four finite singular points and the local chart $U_1$ has no singular
point we obtain the phase portraits $32-36$ in Figure \ref{pppp_2}.
For phase portraits $32$ and $33$ of Figure \ref{pppp_2} we consider the straight line
$x=z_4^1$ where the finite singularity $z_4$ is $z_4=(z_4^1,z_4^2)$, and apply
Lemma \ref{lemma2} to see that they are not realizable (see Figure \ref{32}).
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig17} % lemma2_fig.eps}
\caption{The straight line $x=z_4^1=-(\beta_2-\sqrt{\beta_2^2-4 \alpha_2})/2$ in phase
portrait 32 of Figure \ref{pppp_2}.}
\label{32}
\end{figure}
Moreover the phase portraits $35$ and $36$ are topologically equivalent to the phase portrait
$34$ which is the only realizable phase portrait for this case and it is represented by PL.1.8
in Figure \ref{pp_2}. The values of the parameters that realize this system can be found
in Table \ref{tabela}.
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig18} % pppp_3.eps}
\caption{Phase portraits $41-50$ corresponds to phase portraits of system (P.1) when
$\gamma_2=1$ and $\beta_2^2-4 \alpha_2>0$; Phase portraits $51-60$ corresponds to phase
portraits of system (P.1) when $\gamma_2=1$ and $\beta_2^2-4 \alpha_2=0$.}
\label{pppp_3}
\end{figure}
For $\beta_2^2-4 \alpha_2>0$ we consider the cases with three finite singular points.
When $z_1=z_2$ the origin is a saddle-node and there are ten possible phase portraits,
namely 37--40 in Figure \ref{pppp_2} and 41--46 in Figure \ref{pppp_3}.
But since the nodal sector of the saddle node must have its orbits tangent to its
separatrix, the phase portraits $37$ and $38$ in Figure \ref{pppp_2} are not realizable.
In other words the separatrices of the saddle-node $z_1$ must be on the invariant parabola.
With the same argument the phase portraits 41, 42, 45 and 46 of Figure \ref{pppp_3}
also are not realizable. So when $z_1=z_2$ the realizable phase portraits are 39, 40, 43 and 44
of Figure \ref{pppp_3}, corresponding to PL.1.9, PL.1.10, PL.1.11 and PL.1.12
in Figure \ref{pp_2}, respectively. The values of the parameters that realize these
systems can be found in Table \ref{tabela}.
When there are three finite singularities with $z_1=z_4$ then by continuity we have the
phase portraits 47--50 of Figure \ref{pppp_3}. All these for phase portraits are realizable
and correspond, to PL.1.13, PL.1.14, PL.1.15 and PL.1.16 in Figure \ref{pp_2}, respectively.
The values of the parameters that realize these systems can be found in Table \ref{tabela}
For $\beta_2^2-4 \alpha_2=0$ there is another case with three finite singularities that
correspond to the case $z_3=z_4$. Here we can have ten phase portraits, given by $51-60$
in Figure \ref{pppp_3}. The phase portraits $51, 52$ and $55$ are realizable and corresponds,
respectively, to PL.1.17, PL.1.18 and PL.1.19 in Figure \ref{pp_2}.
The values of the parameters that realize these systems can be found in Table \ref{tabela}.
The phase portraits $53$ and $54$ are not realizable. The idea again is to use Lemma \ref{lemma2}
with the straight line joining the origin of the local chart $U_3$ to the singular point
$u_2$ of the local chart $U_1$. By Figure \ref{5354} and this lemma the phase portraits
$53$ and $54$ are not realizable.
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig19} % 5354.eps}
\caption{The straight line connecting the origin of the local chart $U_3$ with the singular
point $u_2$ of the local chart $U_1$ in phase portrait 53 of Figure \ref{pppp_3}.}
\label{5354}
\end{figure}
Considering the phase portraits $56$ and $57$ we will show that they are not realizable.
Take the straight line passing through the origin of the local chart $U_1$ and the infinite
singular point $u_1=u_2$ (see Figure \ref{56}). The contact points on this straight line
contradicts Lemma \ref{lemma2} so the phase portraits $56$ and $57$ are not realizable.
About the phase portraits $58$ and $59$, considering the straight line passing through
the points $z_1$ and $z_3$ we have Figure \ref{58} that is a contradiction with
Lemma \ref{lemma1}. So they are not realizable. The phase portrait $60$ is topologically
equivalent to $50$ of Figure \ref{pppp_3}.
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig20} %56.eps
\caption{The straight line connecting the origin of the local chart $U_3$ with the singular
point $u_1=u_2$ of the local chart $U_1$ in phase portrait 56 of Figure \ref{pppp_3}.}
\label{56}
\end{figure}
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig21} % 58.eps
\caption{The straight line passing through the points $z_1$ and $z_3$ in phase portrait 58
of Figure \ref{pppp_3}.}
\label{58}
\end{figure}
If $z_3=z_4$ and $z_1=z_2$ we have the phase portraits 61, 62 and 63 of Figure \ref{pppp_4}.
But using the straight line joining $z_1$ and $z_3$ as done in Figure \ref{58} and applying
Lemma \ref{lemma1} we see that $61$ and $62$ are not realizable. The phase portrait $63$
is realizable and corresponds to PL.1.20 in Figure \ref{pp_2}. The values of the parameters
that realize this system can be found in Table \ref{tabela}.
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig22} % pppp_4.eps}
\caption{Phase portraits $61-63$ corresponds to phase portraits of system (P.1) when
$\gamma_2=1$ and $\beta_2^2-4 \alpha_2=0$; Phase portraits $64-72$ corresponds to phase
portraits of system (P.1) when $\gamma_2=1$ and $\beta_2^2-4 \alpha_2<0$.}
\label{pppp_4}
\end{figure}
For $\beta_2^2-4 \alpha_2<0$ the points $z_3$ and $z_4$ are complex. The possible phase
portraits are described by $64-72$ of Figure \ref{pppp_4}. The phase portraits 64, 65, 68 and 71
are realizable and corresponds, respectively, to PL.1.21, PL.1.22, PL.1.23 and PL.1.24
of Figure \ref{pp_2}. The values of the parameters that realize these systems can be
found in Table \ref{tabela}. To prove that the phase portraits 66, 67, 69 and 70 are not
realizable, it is enough to consider the straight line passing through the origin of
the local chart $U_3$ and the infinity singularity $u_1=u_2$ of the local chart $U_1$
(see Figure \ref{66}). This straight line generates a contradition with Lemma \ref{lemma2}
so the phase portraits 66, 67, 69 and 70 are not realizable.
\begin{figure}[http]
\centering
\includegraphics[scale=0.6]{fig23} % 66.eps
\caption{The straight line connecting the origin of the local chart $U_3$ with the singular poin $u_1=u_2$ in the local chart $U_1$ in phase portrait 66 of Figure \ref{pppp_4}.}
\label{66}
\end{figure}
To end the case $\gamma_2=1$ we consider the case where there is only one finite singular point.
Using \cite[Theorem 2.19]{Llibre} we can see that the point is a saddle, which generates phase
portrait 72 of Figure \ref{pppp_4} which corresponds to phase portrait PL.1.25
of Figure \ref{pp_3}. The values of the parameters that realize this system can be found
in Table \ref{tabela}.
Now we consider the case $\gamma_2=0$. The system is
\begin{equation*} % \label{p1proof}
\dot{x}=x (\alpha_2+\beta_2 x),\quad
\dot{y}=\alpha_1( y-x^2)+ 2x( \alpha_2 x+\beta_2 y).
\end{equation*}
When $\alpha_1=0$ such system has a common factor so assume $\alpha_1=1$. By the change
$x = -X, y= Y$ it is enough to consider the case $\beta_2 \geq 0$.
Assuming $\beta_2>0$. In the finite part the points $z_1=(0,0)$ and
$z_2=\left(-\alpha_2/\beta_2,(\alpha_2/\beta_2)^2\right)$ are the singular points and the
system has an additional invariant straight line, given by $f_3=x+\alpha_2/\beta_2=0$.
Defining $l_1=\alpha_2$ and $l_2=1-2\alpha_2$ the eigenvalues associated to $z_1$ are
$1$ and $l_1$, while the eigenvalues associated to $z_2$ are $-l_1$ and $l_2$.
We assume $l_2 \neq 0$ (otherwise such system has a common factor and it is equivalent
to a linear system).
In the local chart $U_1$ the unique singular point is $u_1=(l_2/\beta_2,\, 0)$ and it is a
saddle. In the local chart $U_2$ the compactified system is
\begin{equation*}
\dot{u}=u ((1-2 \alpha_2) u^2+(\alpha_2-1) v-\beta_2 u),\quad
\dot{v}=v ((1-2 \alpha_2) u^2-2 \beta_2 u-v).
\end{equation*}
The origin $(0,0)$ is a linearly zero singularity. Doing the blow up $u= UV,\, v = V$
and rescaling by $V$ we obtain the system
\begin{equation*}
\dot{U}=U (\alpha_2+\beta_2 U),\quad
\dot{V}=V ((1-2 \alpha_2 U^2 V)-2 \beta_2 U-1).
\end{equation*}
When $V=0$ the singularities are $\overline{u}_1=(0,0)$ and
$\overline{u}_2=(-\alpha_2/\beta_2,0)$. The eigenvalues associated to $\overline{u}_1$ are
$-1$ and $l_1$ while the eigenvalues of $\overline{u}_2$ are $-l_1$ and $-l_2$.
The blowing down process is described in Figure \ref{bd_p_1} (1)-(4) according to the signs
of $l_1$ and $l_2$.
When $\beta_2=0$ the point $z_1$ is the unique finite singular point, being a saddle
or an unstable node depending on the sign of $l_1$. In the local chart $U_1$ there is no
singular point and the origin $(0,0)$ of $U_2$ is linearly zero. To study such point we
apply \emph{blow ups}, in Figure \ref{bd_p_1} is described the blowing down (5) and (6).
Summarizing the study done previously we obtain the local behaviour at origin of $U_2$:
\begin{enumerate}
\item $\beta_2>0, \,l_1>0$ and $l_2>0$: the origin of $U_2$ has two elliptic sectors;
\item $\beta_2>0, l_1>0$ and $l_2<0$: the origin of $U_2$ has two hyperbolic sectors;
\item $\beta_2>0, l_1<0$ and $l_2>0$: the origin of $U_2$ has two elliptic sectors;
\item $\beta_2>0, l_1=0$ and $l_2>0$: the origin of $U_2$ has two elliptic sectors.
\item $\beta_2=0, l_1>0$: the origin of $U_2$ has two hyperbolic sectors;
\item $\beta_2=0, l_1<0$: the origin of $U_2$ has two elliptic sectors.
\end{enumerate}
\begin{figure}[http]\centering
\includegraphics[scale=0.68]{fig24} % bd_p1.eps}
\caption{\emph{Blow down} of system (P.1) when $\gamma_2=0$.}
\label{bd_p_1}
\end{figure}
By continuity and the above analysis we conclude that the case $(3)$ is topologically
equivalent to case (1) and the cases (1), (2), (4), (5) and (6) correspond, respectively,
to the phase portraits PL.1.26, PL.1.27, PL.1.28, PL.1.29 and PL.1.30 of Figure \ref{pp_3}.
Table \ref{tabelaRPL} has the values of the parameters that realizes the phase portraits
of system (P.1)
\begin{table}[http]
\caption{Table of values for the parameters of system (P.1).}
\label{tabela}
\centering
\begin{tabular}{|l|c|c|r|r|}
\hline
& $\gamma_2$ &$ \beta_2$ & $\alpha_2$ & $\alpha_1$\\
\hline
PL.1.1 & 1&1&1/8&1\\
PL.1.2 & 1 & 1 & 1/16 & 1/16\\
PL.1.3& \multicolumn{4}{|c|}{by continuity}\\
PL.1.4 & 1 & 1 & 1/16 & 1/150\\
PL.1.5 & 1 & 1/2 & 3/64 & 1/32\\
PL.1.6& \multicolumn{4}{|c|}{by continuity}\\
PL.1.7 & 1 & 1&-3/8& -1 \\
PL.1.8 & 1 & 1 & 3/16 & 1/16\\
PL.1.9 & 1 & 1 & 1/16 & 0\\
PL.1.10 &1 & 1 &-1 & 0\\
PL.1.11 &1 & 1 &1/18 & 0\\
PL.1.12 &1 & 1 &3/16& 0\\
PL.1.13 &1 & 1 &0& 1\\
PL.1.14 &1 & 1 &0& -1/8\\
PL.1.15 &1 & 1 &0& -1/4\\
PL.1.16 &1 & 1 &0& -1\\
PL.1.17 &1 & 1 &1/4& 1\\
PL.1.18 &1 & 1 &1/4& 3/8\\
PL.1.19 &1 & 1 &1/4& 1/4\\
PL.1.20 &1 & 1 &1/4& 0\\
PL.1.21 &1 & 1 &2& 5\\
PL.1.22 &1 & 3 &4& 6\\
PL.1.23 &1 & 1 &9/8& 2\\
PL.1.24 &1 &1 &2 & 13/4\\
PL.1.25 & 1 & 1 & 2& 0\\
PL.1.26 & 0 & 1 & 1/4& 1\\
PL.1.27 & 0 & 1 & 3/2& 1\\
PL.1.28 & 0 & 1 & 0& 1\\
PL.1.29 & 0 & 0 & 1& 1\\
PL.1.30 & 0 & 0 & -1& 1\\
\hline
\end{tabular}
\end{table}
\begin{figure}[http]\centering
\includegraphics[scale=01]{fig25} % teonilpotente.eps
\caption{Local phase portraits }
\label{teonilpotente}
\end{figure}
System (P.2) with $c\neq 0$ has a Darboux invariant if $\gamma_2 \neq0$, and it can be
written as \begin{equation*}
\dot{x}=\beta_1 (x^2+c)+\gamma_2 x(y+c),\quad \dot{y}=2 (y+c) (\beta_1 x+\gamma_2 y).
\end{equation*}
Note that if $\beta_1=0$ such system has a common factor so we can assume $\beta_1=1$.
Applying the change of coordinates $x = -X, y = Y$ and rescaling the time we can assume
$\gamma_2>0$.
If $c<0$ the system has three finite singular points $z_1=(-1/\gamma_2,1/\gamma_2^2)$,
$z_2=(-\sqrt{-c}, -c)$ and $z_3=(\sqrt{-c}, -c)$. Otherwise, only $z_1$.
Defining $l_1=c \neq 0$ and $l_2=1+c\, \gamma_2^2 $ the eigenvalues associated to $z_1$
are $2\gamma_2 l_1$ and $l_2/\gamma_2$, the eigenvalues associated to $z_2$ are $-2\sqrt{-c}$
and $-2(\gamma_2 \,c+\sqrt{-c})$; the eigenvalues associated to $z_3$ are $2\sqrt{-c}$ and
$-2(\gamma_2 \,c-\sqrt{-c})$. So when $c<0$ the point $z_3$ exists and it is an unstable node.
In the local chart $U_1$ we have two singular points $u_1=(0,0)$ being a hyperbolic saddle
and $u_2=(-1/\gamma_2,0)$ being a saddle-node. In the local chart $U_2$ the origin is a
stable node.
When $l_2=0$ then $z_1=z_3$ is a semi-hyperbolic node and the infinity part does not change.
Note that $z_1$ is a saddle-node in this case.
So by continuity and the reasoning above, if $c>0$ we have phase portrait PL.2.1 of
Figure \ref{pp_3} which is realizable with $c=\gamma_2=1$. When $c<0$ and $l_2 \neq 0$ the
system has two possible phase portraits, also described in Figure \ref{pp_3}:
PL.2.2 (realizable with $c=-1/2$ and $\gamma_2=1$) and PL.2.3 (realizable with $c=-2$ and
$\gamma_2=1$).
Finally if $c<0$ and $l_2=0$, we see that the line $y + c = 0$ is one of the separatix of
the saddle-node. So the only possible phase picture is PL.2.4 (realizable with $c=-1$
and $\gamma_2=1$).
Now we study the global phase portraits of systems (P.2) when $c=0$ and they have a
Darboux invariant. The differential system is
\begin{equation*}
\dot{x}=-\beta_1(y-x^2)+\beta_2 y+\alpha_2 x,\quad \dot{y}=2y(\beta_2 x+\alpha_2).
\end{equation*}
Since $\alpha_2 \neq 0$ we take $\alpha_2=1$. Moreover doing the change of coordinates
$x = -X, y = Y$ we can assume $\beta_2 \geq 0$. The system has at most three finite
singular points, namely, $z_1=(0,0)$ and $z_2=(-1/\beta_1,0)$ and $z_3=(-1/\beta_2,1/\beta_2^2)$.
The point $z_1$ has eigenvalues $2$ and $1$, so it is an unstable node. On the other hand the
topological type of $z_2$ and $z_3$ depends on the numbers $l_1\doteq \beta_1$ and
$l_2\doteq \beta_1-\beta_2 \neq 0$. Indeed the point $z_2$ has eigenvalues $-1$ and
$2l_2/l_1$ and $z_3$ has the eigenvalues $-1$ and $-2l_2/l_1$.
In the local chart $U_1$ the system has $u_1=(0,0)$ as a singularity with eigenvalues $-l_1$
and $-l_3$, where $l_3 = \beta_1-2\beta_2$.
In the local chart $U_2$ the compactified system has the origin as a \emph{nilpotent}
singularity. This mean that the linear part of the system, evaluated in $(0,0)$, is not
null but their eigenvalues are both equal to zero. To classify this type of singular
point we use Theorem 3.5 of \cite{Llibre}. This result use two functions, $F(u)=a_M u^M+o(u^M)$
and $G(u)=b_Nu^N+o(u^N)$, defined from the differential system. In short the caracterization
is done using $a_M, b_N$ and the natural numbers $M, N$.
For the compactified system in the local chart $U_2$ these functions are
\begin{equation*}
G(u)=-\frac{2(\beta_2-3\beta_2)}{l_2}\, u+ \frac{ 5l_3}{l_2^2}\, u^2,\quad
F(u)= \frac{2\beta_2l_3}{l_2^2}\,u^3+\frac{2l_3^2}{l_2^3}\,u^4.
\end{equation*}
So when $l_3>0$ the origin $(0,0)$ is a saddle as in (b) of Figure \ref{teonilpotente}.
If $l_3<0$ the origins consists of one hyperbolic and one elliptic sector as in (a)
of Figure \ref{teonilpotente}.
By continuity, when $l_1>0$ and $l_3>0$ we have the phase portrait PL.2.5 of Figure \ref{pp_3}
(realizable with $\beta_1=4$ and $\beta_2=1$). If $l_3<0$ we have the phase portraits PL.2.6
(realizable with $\beta_1=3/2$ and $\beta_2=1$) and PL.2.7 (realizable with $\beta_1=1/2$
and $\beta_2=1$) of Figure \ref{pp_3}. Now if $l_1<0$ the only possibility is $l_3<0$
and we have the phase portrait PL.2.8 (realizable with $\beta_1=-1$ and $\beta_2=1$) of
Figure \ref{pp_3}.
If $l_1=0$ the point $z_2$ goes to the infinity and collide with $u_1$ becoming a saddle-node.
Moreover $l_1=0$ implies $l_3<0$, so the origin of $U_2$ has a hyperbolic and one elliptic
sector. This case corresponds to phase portrait PL.$2.9$ of Figure \ref{pp_3}, realizable
with $\beta_1=0$ and $\beta_2=1$.
If $\beta_2=0$ the point $z_3$ goes to the infinity and collide with the origin of $U_2$
becoming $(0,0)$ a nilpotent saddle-node as $(c)$ or $(d)$ in Figure \ref{teonilpotente}.
Moreover the only possible phase portrait is given by PL.2.10 of Figure \ref{pp_3},
realizable with $\beta_1=1$ and $\beta_2=0$).
Finally when $l_3=0$ then the infinity if filled of singular points, without special
singularities and the corresponding phase portrait is PL.2.11 of Figure \ref{pp_3}
(realizable with $\beta_1=2$ and $\beta_2=1$).
\end{proof}
\begin{proposition} \label{LV}
Each real planar polynomial differential system with two invariant real lines that
intersect at a single point and a third invariant straight line having a Darboux invariant
can be written, after an affine change of coordinates, as
\begin{itemize}
\item [(i)] {\rm (LV.1)} with $\alpha(q-\beta) \neq$ and Darboux invariant
\begin{equation*}
I_{10}(x,y,t)=e^{\alpha (q-\beta )\,t} y^{\beta } x^{\beta -q+r} (y-x)^{-(\beta +r)},
\end{equation*}
\item [(ii)] {\rm (LV.2)} with $c=q=0$, $p \neq 0$ and Darboux invariant
\begin{equation*}
I_{11}(x,y,t)= e^{-p t}x y^{-r},
\end{equation*}
\item [(iii)] {\rm (LV.2)} with $c =-1 $ and Darboux invariant
\begin{equation*}
I_{12}(x,y,t)=e^{t} y\, (y-1)^{-1},
\end{equation*}
\item [(iv)] {\rm (LV.3)} with $\alpha=-(\beta+1)$, $c\,\beta \neq 0$ and Darboux invariant
\begin{equation*}
I_{13}(x,y,t)=e^{-c\,\beta \,t} y\,(y+a x+c)^{-1}.
\end{equation*}
\end{itemize}
Moreover there are 27 non-equivalent phase portraits in the Poincar\'e disc.
They are in Figure \ref{pp_4}.
\end{proposition}
\begin{proof}
Let $f_1=x=0$, $f_2=y=0$ be the two real straight lines intersecting in a point.
Considering system (LV.1) the third line is $f_3=y-x$ and the cofactors associated to
$f_1, f_2$ and $f_3$ are, respectivelly, $k_1=\alpha +r y+\beta x$,
$k_2=\alpha +y (\beta -q+r)+q x$ and $k_3=\alpha +y (\beta -q+r)+\beta x$.
One solution for equation $\lambda_1 k_1+\lambda_2 k_2+s=0$ is
\[
\lambda_2= \frac{\beta\lambda_1 }{\beta -q+r}, \quad
\lambda_3= -\frac{(\beta +r)\lambda_1}{\beta -q+r}, \quad
s= \frac{\alpha (q-\beta )\lambda_1}{\beta -q+r},
\]
Taking $\lambda_1=\beta -q+r$ we obtain the Darboux invariant
\[
I_{10}(x,y,t)=e^{\alpha (q-\beta )\,t} y^{\beta } x^{\beta -q+r} (y-x)^{-(\beta +r)}.
\]
Now we analyze system (LV.2) that has $f_3=y+c$ as the third invariant straight
line(remember that $c=0$ or $c=-1$). Here the cofactors are $k_1=p+q x+r y$, $k_2=y+c$ and
$k_3=y$. If $c=0$ then equation \eqref{igualdade de darboux} has only one the solution
\begin{equation*}
q= 0, \quad \lambda_3= -r\lambda_1-\lambda_2,\quad s= -p\lambda_1.
\end{equation*}
Taking $\lambda_1=1$ we obtain the Darboux invariant
\[
I_{11}(x,y,t)= e^{-p t}x y^{-r}.
\]
Otherwise if $c=-1$ then the more general solution is
\begin{equation*}
\lambda_1= 0,\,\lambda_3= -\lambda_2,\, s= \lambda_2.
\end{equation*}
Taking $\lambda_2=1$ we obtain the Darboux invariant
\[
I_{12}(x,y,t)= e^{t}y(y-1)^{-1}.
\]
The last case to be considered is system $(LV.3)$ that has $f_3=y+a x+c=0$ as the third
straight line. The cofactors are $k_1=-\alpha(y+a x+c)-y,\, k_2= \beta(y+a x+c)+a x$ and
$k_3=\beta y-a \alpha x$. Solving equation \eqref{igualdade de darboux} we obtain the solution
\begin{equation*}
\alpha = -(\beta+1), \quad \lambda_2 = -\lambda_1-\lambda_2, \quad
s = -c(\lambda_1+\beta(\lambda_1+\lambda_2)).
\end{equation*}
Taking $\lambda_1=0$ and $\lambda_2=1$ then we obtain the Darboux invariant
\[
I_{13}(x,y,t)=e^{-c\,\beta \,t} y\,(y+a x+c)^{-1}.
\]
We begin the study of the global phase portraits with systems (LV.1)
when they have a Darboux invariant. Remember that if system $(LV.1)$ has a Darboux
invariant then $\beta-q \neq 0$ and $\alpha \neq 0$ so we can take $\alpha=1$ getting
\begin{equation} \label{lv3}
\dot{x}=x(1+\beta x+r y),\quad
\dot{y}=y (1+q x+(\beta-q+r) y).
\end{equation}
Define $l_1=(\beta-q)/(\beta-q+r), \, l_2=(\beta-q)/\beta$ and $l_3=(\beta-q)/(\beta+r)$.
The finite part presents at most four singularities
\begin{itemize}
\item $z_1=(0,0)$ with eigenvalues both equal to $1$;
\item $z_2=(0,-1/(\beta-q+r))$ with eigenvalues $-1$ and $l_1$;
\item $z_3=(-1/\beta,0)$ with eigenvalues $-1$ and $l_2$;
\item $z_4=(-1/(\beta+r),\,-1/(\beta+r))$ with eigenvalues $-1$ and $-l_3$.
\end{itemize}
In the local chart $U_1$ the compactified system has two singular points, being
$u_1=(0,0)$ with eigenvalues $-\beta$ and $-(\beta-q)$ and $u_2=(1,0)$ with eigenvalues
$\beta-q$ and $-(\beta+r)$. Moreover in the local chart $U_2$ the origin $(0,0)$ is a
singular point with eigenvalues $-(\beta-q)$ and $-(\beta-q+r)$.
Thus when one of the finite singularities goes to infinity, it collides with $u_1$, $u_2$,
or the origin of the local chart $U_2$.
When $l_1, l_2$ and $l_3$ are non-zero, the combinations between their signs generate
the possible phase portraits of system \eqref{lv3}. There are exactly three possible phase
portraits, all of them described in Figure \ref{pp_4}: LVL.1.1, realizable for
$\beta=1$, $q=r=0$; LVL.1.2, realizable for $\beta=1$, $q=r=-2$; LVL.1.3, realizable
for $\beta=1$, $q=-r=3/4$.
Now we consider the case $\beta=-r\neq 0$. Here only the point $z_4$ goes to the infinity
and collides with $u_2$ making it a semi hyperbolic saddle-node. There are two possible phase
portraits, given by LVL.1.4 of Figure \ref{pp_4} (realizable with $\beta=1$, $q=r=-1$)
and LVL.1.5 of Figure \ref{pp_4} (realizable with $\beta=2$, $q=1$, $r=-2$).
The cases where $z_2$ or $z_3$ goes to the infinity generate phase portraits equivalent
to the previous ones.
Finally when two finite singular points go to the infinity (for example when $\beta=-r$ and
$q=0$), then there is only one phase portrait, given by LVL.1.6 of Figure \ref{pp_4}.
This last phase portrait is realizable for $\beta=1$, $q=0$ and $r=-1$.
Now we consider the systems (LV.2) when they have a Darboux invariant we split in two cases.
First we consider the case $c=-1$, when the system is given by
\[
\dot{x}=x(p+q x+r y),\quad \dot{y}=y (y-1).
\]
If $q \neq 0$ unless of the change $x = X/q$ we can assume $q=1$. Considering $q=1$ and
defining $l_1=p$, $l_2=-(p+r)$ and $l_3=r-1$ the system has at most four finite singular points,
namely
\begin{itemize}
\item $z_1=(0,0)$ with eigenvalues $-1$ and $l_1$;
\item $z_2=(0,1)$ with eigenvalues $1$ and $-l_2$;
\item $z_3=(-p,0)$ with eigenvalues $-1$ and $-l_1$;
\item $z_4=(-p-r,1)$ with eigenvalues $1$ and $l_2$.
\end{itemize}
In the local chart $U_2$ the origin $(0,0)$ is a singularity with eigenvalues $-1$ and $l_3$.
In the local chart $U_1$ the system has two singularities if $l_3 \neq 0$: $u_1=(0,0)$ being a
hyperbolic unstable node and $u_2=(1/l_3,0)$ with eigenvalues $1$ and $1/l_3$.
Hence if $l_3=0$ the point $u_2$ collides with the origin of $U_2$ making it a semi-hyperbolic
singularity of type saddle node. By continuity and using all the possible combinations of the
signs of $l_1, l_2$ and $l_3$ when $q=1$ and $l_3 \neq 0$ we obtain the phase portraits
LVL.2.1--LVL.2.7 of Figure \ref{pp_4}. When $l_3=0$, i.e., $r=1$ has three possible phase
portraits: LVL.2.8, LVL.2.9 and LVL.2.10 of Figure \ref{pp_4}.
The values of the parameters that realize these systems can be found in Table \ref{tablelv}.
Now it remains to study the case $q=0$. Note that since the system cannot have common
factors it follows that $l_1$ and $l_2$ are different from zero. When $q=0$ both the finite
part and the analyzes in the local chart $U_2$ remain almost the same.
The only difference in the finite part is that the singularities $z_3$ and $z_4$ go to
infinity. However in the local chart $U_1$ the compactified system is
\begin{equation*}
\dot{u}=-u ((r-1) u+(p+1)v),\quad \dot{v}=-v (p v+r u).
\end{equation*}
So the origin is a linearly zero singular point if $l_3 \neq 0$ and we apply the
\emph{blow up} doing the change of coordinates $u = U, v = UW$.
The new system is
\begin{equation*}
\dot{U}=-U^2 ((p+1) W+r-1), \quad
\dot{W}=UW (W-1).
\end{equation*}
After eliminating the common factor $U$ it remains two singular points on $U=0$:
$\overline{u_1}=(0,0)$ with eigenvalues $-1$ and $-l_3$, and $\overline{u_2}=(0,1)$
with eigenvalues $1$ and $l_2$. Hence they are hyperbolic points and doing the
\emph{blow down} the origin of $U_2$ has (for $l_3 \neq 0$)
\begin{itemize}
\item two elliptic sectors if $\overline{u_1}$ is a saddle and $\overline{u_2}$ is a
unstable node. This case corresponds to phase portrait LVL.2.11 of Figure \ref{pp_4};
\item two elliptic sectors if $\overline{u_1}$ is a stable node and $\overline{u_2}$ is a saddle. This case corresponds to phase portrait LVL.$2.12$ of Figure \ref{pp_4};
\item two parabolic sectors if $\overline{u_1}$ and $\overline{u_2}$ are both saddles and
there is a saddle and a node as singular finite points. This case corresponds to phase
portrait LVL.2.13 of Figure \ref{pp_4};
\item two parabolic sectors if $\overline{u_1}$ and $\overline{u_2}$ are both saddles
and there are two nodes as singular finite points. This case corresponds to phase portrait
LVL.2.14 of Figure \ref{pp_4};
\item six parabolic sectors if $\overline{u_1}$ and $\overline{u_2}$ are both saddles
and there are two nodes as singular finite points. This case corresponds to phase
portrait LVL.2.15 of Figure \ref{pp_4}.
\end{itemize}
The last possibility when $c=-1$ is $q=0$ and $l_3=0$. But when this happens the system
has the infinity line $v=0$ filled up of singular points. After eliminating the common
factor $v$, in the local chart $U_1$ the point $u_1=(0,0)$ is a singular point,
with eigenvalues $-l_1$ and $l_2$. In the local chart $U_2$, After eliminating the common
factor $v$, the origin is a singularity. By continuity the possible phase portraits are LVL.2.16
and LVL.2.17 of Figure \ref{pp_4}. In Table \ref{tablelv} we put the values of the parameters
that realizes each one of the phase portraits described in Figure \ref{pp_4}.
\begin{table}
\caption{Table of values for the parameters of system (LV.2) when $c=-1$.}
\label{tablelv}
\centering
\begin{tabular}{|l|c|r|r|}
\hline
&$q$& $r$ &$p$ \\
LVL.2.1 & 1 & -1&1/2\\
LVL.2.2 & 1 & 2&1\\
LVL.2.3 & 1 & -1&2\\
LVL.2.4 & 1 & -1&1\\
LVL.2.5 & 1 & 2&-2\\
LVL.2.6 & 1& 1/2&-1/2\\
LVL.2.7 & 1 & 0&0\\
LVL.2.8 & 1 & 1&1\\
LVL.2.9 & 1 & 1&-1/2\\
LVL.2.10 & 1 & 1&-1\\
LVL.2.11 & 0 & -2&1\\
LVL.2.12 & 0 & 2&1\\
LVL.2.13 & 0 & 0&1\\
LVL.2.14 & 0& 2&-1\\
LVL.2.15 & 0 & 3/4&-1/4\\
LVL.2.16 & 0& 1&1\\
LVL.2.17 & 0& 1&-1/2\\
\hline
\end{tabular}
\end{table}
Finally when $c=0$ we obtain the differential system
\begin{equation}
\label{pdl}
\dot{x}=x^2, \quad
\dot{y}=y (p+r x),
\end{equation}
with $p \neq 0$. So we can take $p=1$ and the system becomes a particular case of system
(DL) of Theorem \ref{teo1}. The global phase portraits of this system will be done in the
proof of Proposition \ref{DL} and the correponding phase portraits of system \eqref{pdl}
are described by DL.1, DL.2 and DL.3 of Figure \ref{pp_5}.
To complete the proof of Proposition \ref{LV} we study the global phase portraits of systems
(LV.3). When (LV.3) has a Darboux invariant the parameter $\alpha$ must be equal to
$-(\beta+1)$ so the differential system is
\begin{equation*}
\dot{x}=x(a x+\beta(y+a x+c)+c), \quad
\dot{y}=y (a x+\beta(y+a x+c)).
\end{equation*}
In the finite part there are three singular points, namely $z_1=(0,0), \, z_2=(0,-c)$ and
$z_3=(-c/a,0)$ (remember that $a\,c \neq 0$). Defining $l_1=c\, \beta \neq 0$ and
$l_2=c(\beta+1) \neq 0$, then the eigenvalues of the $z_1$ are $l_1$ and $l_2$;
the eigenvalues of $z_2$ are $c$ and $-l_1$, and the eigenvalues associated to $z_3$ are
$-c$ and $-l_2$.
In the local chart $U_1$ the compactified system becomes
\begin{equation*}
\dot{u}=-cuv, \quad
\dot{v}=-v(c v+\beta (u+cv+a)+a).
\end{equation*}
Hence the line $v=0$ is filled of singular points after eliminating the common factor $v$
there are no singular points. The same happens in the local chart $U_2$.
So by continuity the only possible phase portrait is LVL.3.1 of Figure \ref{pp_4},
which is realizable for $\beta=1$ and $a=c=-1$.
\end{proof}
\begin{proposition} \label{RPL}
Each real planar quadratic differential system with two parallel real invariant straight
lines and a third invariant straight line having a Darboux invariant can be written,
after an affine change of coordinates, as system {\rm (RPL)} and it has the Darboux invariant
\[
I_{14}(x,y,t)= e^{2 t} (x+1)(x-1)^{-1}.
\]
Moreover there are 17 non-equivalent phase portraits in the Poincar\'e disc for this system.
They are described by {\rm RPL.1--RPL.17} in Figure \ref{pp_5}.
\end{proposition}
\begin{proof}
Let $f_1=x+1=0$, $f_2=x-1=0$ and $f_3=y=0$ be the three invariant straight lines.
The cofactors of $f_1,f_2$ and $f_3$ are, respectivelly, $k_1=x-1$, $k_2=x+1$,
$k_3=\alpha +\beta x+\gamma y$. With these cofactors equation \eqref{igualdade de darboux}
with $s \in \mathbb{R} \setminus \{0\}$ has two solutions, namely
\begin{gather*}
s_1=\{\gamma= 0, \, s=2 \lambda_1 +(\beta-\alpha) \lambda_3, \lambda_2
=-(\lambda_1 +\beta \lambda_3)\}\\
s_2=\{s= 2\lambda_1, \, \lambda_2= -\lambda_1, \, \lambda_3= 0\}.\\
\end{gather*}
Since the second solution $s_2$ is more general, we conclude that every quadratic system
that has two real parallel straight lines and a third real straight line as invariant straight
lines also has a Darboux invariant. Taking $\lambda_1=1$ we obtain the invariant
\[I_{14}(x,y,t)=e^{2t} (x+1)(x-1)^{-1}.\]To draw the possible global phase portraits,
remember that the system is
\begin{equation*}
\dot{x}=x^2-1,\quad
\dot{y}=y (\alpha +\beta x+\gamma y).
\end{equation*}
When $\gamma \neq 0$ we can take $\gamma=1$ (indeed, just do the change $x = X, y = Y/\gamma$).
So the system can present at most four finite singularities, namely, $z_1=(-1,0)$,
$z_2=(-1, \beta-\alpha)$, $z_3=(1,0)$ and $z_4=(1,-\beta-\alpha)$.
Define $l_1=\alpha-\beta$ and $l_2=\alpha+\beta$. The eigenvalues associated to $z_1$ are
$-2$ and $l_1$ while the eigenvalues associated to $z_2$ are $-2$ and $-l_1$.
Moreover $z_1=z_2$ when $l_1=0$. Analogously the eigenvalues of $z_3$ are $2$ and $l_2$,
while the eigenvalues associated to $z_4$ are $2$ and $-l_2$, with $z_3=z_4$ when $l_2=0$.
So in the finite part the system can have two, three or four singularities, depending on the
values of $l_1$ and $l_2$.
In the local chart $U_1$ the compactified system has at most two singularities on the infinity
line: $u_1=(0,0)$ and $u_2=(1-\beta,0)$. Defining $l_3=\beta-1$ we see that $u_1=u_2$ when
$l_3=0$ and the topological type of these singularities depends on the sign of $l_3$.
Indeed the eigenvalues associated to $u_1$ are $-1$ and $l_3$ while the associated to $u_2$
are $-1$ and $-l_3$.
In the local chart $U_2$ we just need to check if the origin $(0,0)$ is a singularity,
which is true. It is a node, with the two eigenvalues equal to $-1$.
So considering $\gamma \neq 0$ and combining all the possibilities of the signs of
$l_1, l_2$ and $l_3$ we obtain the phase portraits RPL.1--RPL.10 of Figure \ref{pp_5}.
In Table \ref{tablelv} we put the values of the parameters that realizes each one of the phase
portraits described in Figure \ref{pp_5}.
If $\gamma=0$ then $z_2$ and $z_4$ goes to the infinity and the compactified system in the
local chart $U_2$ becomes
\begin{equation*}
\dot{u}=(1-\beta )u^2-\alpha u v-v^2, \quad
\dot{v}=-v (\beta u+\alpha v).
\end{equation*}
Note that when $l_3=0(\beta=1)$ the line $v=0$ is filled up of singular points, and when
$l_3 \neq 0$ the origin $(0,0)$ is a linearly zero singularity. Considering this case
first and applying the \emph{blow up} $u = U, v = U W$ and dividing by $U$ we obtain the system
\begin{equation}
\label{RPLbd}
\dot{U}= -U (\beta +W^2+\alpha W-1),\quad \dot{W}=W (W-1) (W+1).
\end{equation}
When $U=0$ the singularities of \eqref{RPLbd} are $\overline{u_1}=(0,-1)$ with eigenvalues
$2$ and $l_1$, $\overline{u_2}=(0,0)$ with eigenvalues $-1$ and $-l_3$, and
$\overline{u_3}=(0,1)$ with eigenvalues $2$ and $-l_2$.
After blow-down we obtain the local phase portraits of the origin of $U_2$ which depend on
the signs of $l_1$, $l_2$ and $l_3$. Doing all the combinations the origin of $U_2$ consists of:
\begin{itemize}
\item two elliptic sectors and parabolic sectors, see phase portraits RPL.11 and RPL.12
of Figure \ref{pp_5};
\item two hyperbolic sectors and parabolic sectors, see phase portraits RPL.13 and RPL.14
of Figure \ref{pp_5};
\item six hyperbolic sectors, see phase portrait RPL.15 of Figure \ref{pp_5}.
\end{itemize}
Finally if we consider $\beta=1$ and after eliminating the common factor $v$ the origin of
the local chart $U_2$ is either a hyperbolic node or a hyperbolic saddle, described respectively
by the phase portraits RPL.16 and RPL.17 of Figure \ref{pp_5}. The Table \ref{tabelaRPL}
has the values of the parameters that realizes the phase portraits of Figure \ref{pp_5}.
\begin{table}
\caption{Table of values for the parameters of system (RPL).}
\label{tabelaRPL}
\centering
\begin{tabular}{|l|r|r|c|}
\hline
& $\alpha$ &$ \beta$ & $\gamma$\\
RPL.1 & -5/4&1/4&1\\
RPL.2 & 0&-1&1\\
RPL.3 & -3&2&1\\
RPL.4 & -2&1&1\\
RPL.5 & 0&1&1\\
RPL.6 & -1/2&1/2&1\\
RPL.7 & 1/2&-1/2&1\\
RPL.8 & -2&2&1\\
RPL.9 & -1&1&1\\
RPL.10 & 0&0&1\\
RPL.11 & -3&2&0\\
RPL.12 & 0&-1&0\\
RPL.13 & -1&0&0\\
RPL.14 & -1&2&0\\
RPL.15 & -1/4&3/4&0\\
RPL.16 & -2&1&0\\
RPL.17 & 0&1&0\\
\hline
\end{tabular}
\end{table}
\end{proof}
\begin{proposition} \label{DL}
Each real planar quadratic differential system with a double real invariant straight line and
a third invariant straight line having a Darboux invariant can be written, after an affine
change of coordinates, as system {\rm (DL)}, with $\gamma=0$ and $\alpha \neq 0$, and
the Darboux invariant is
\[
I_{15}(x,y,t)= e^{-\alpha \,t} y x^{-\beta }.
\]
Moreover there are 3 non-equivalent phase portraits in the Poincar\'e disc for this systems.
They are described by {\rm DL.1--DL.3} in Figure \ref{pp_5}.
\end{proposition}
\begin{proof}
Let $f_1=x=0$ be the double real invariant straight line. By the proof of Proposition
\ref{prop1} we know that the second invariant straight line is $f_2=y=0$. The cofactors
of $f_1$ and $f_2$ are, respectively, $k_1=x, k_2=\alpha+\beta x+\gamma y$.
Equation \eqref{igualdade de darboux} with $s \in \mathbb{R} \setminus \{0\}$ has only one solution
$ \gamma= 0$, $s= -\alpha \lambda_2$, $\lambda_1= -\beta \lambda_2 $.
Taking $\lambda_2=1$ and using this solution we obtain
\begin{equation*}
\dot{x}=x^2,\quad
\dot{y}=y (\alpha +\beta x),
\end{equation*}
with Darboux invariant
$ I_{15}(x,y,t)=e^{-\alpha \,t}y x^{-\beta }$.
To study the global phase portraits of systems (DL), since $\alpha \neq 0$ we can take
$\alpha=1$. The origin of the system is the only finite singularity, which is a saddle-node.
For the infinity singularities we assume first that $\beta-1 \neq 0$.
In the local chart $U_1$ the origin is a saddle if $\beta-1>0$, and a stable node if
$\beta-1<0$. In the chart $U_2$ the system becomes
\begin{equation*}
\dot{u}=-u ((\beta-1)u+ v), \quad
\dot{v}-v (\beta u + v),
\end{equation*}
and the origin is a linearly zero singularity. Applying the \emph{blow up} $u = U$, $v = U W$
we obtain the system
\begin{equation*}
\dot{U}=-U^2(\beta-1+W),\quad
\dot{W}=-UW,
\end{equation*}
which after eliminating the common factor $U$ has the origin as only singular point.
If $\beta-1>0$ the origin is a hyperbolic stable node and if $\beta-1<0$ the origin is a saddle.
After \emph{blow down} we obtain the local phase portraits of the origin of $U_2$ which depend
on $\beta$. When $\beta-1>0$ the origin has two elliptic sectors and parabolic sectors,
see phase portrait DL.1 of Figure \ref{pp_5}. If $\beta-1<0$ then there are two hyperbolic
sectors and parabolic ones, see phase portrait DL.$2$ of Figure \ref{pp_5}.
When $\beta=1$ the infinity is filled up of singular points and in the local chart $U_2$
the origin is a stable node. The phase portrait is described by DL.3
of Figure \ref{pp_5}.
\end{proof}
\begin{proposition} \label{CPL}
Each real planar quadratic differential system with two parallel complex invariant straight
lines and a third invariant straight line having a Darboux invariant can be written,
after an affine change of coordinates, as system {\rm (CPL)}. A Darboux invariant is given by
\[
I_{16}(x,y,t)=e^t e^{\arctan (1/x)}
\]
Moreover there are 7 non-equivalent phase portraits in the Poincar\'e disc for this system.
They are described by {\rm CPL.1--CPL.7} in Figure \ref{pp_6}.
\end{proposition}
\begin{proof}
Let $f_1=x+i=0$, $f_2=x-i=0$ be the two complex parallel straight lines.
By the proof of Proposition \ref{prop1} we know that the third invariant straight line
is $f_3=y=0$. The cofactors of $f_1,f_2$ and $f_3$ are, respectively,
$k_1=x-i, k_2=x+i, k_3=\alpha+\beta x+\gamma y$. The equation \eqref{igualdade de darboux}
with $s \in \mathbb{R} \setminus \{0\}$ has two solutions, namely
\begin{gather*}
s_1=\{\gamma= 0, \, s= i(2 \lambda_1+(\beta+i \alpha) \lambda_3),
\, \lambda_2= -\beta \lambda_3-\lambda_1\}\\
s_2=\{s= 2i\lambda_1, \, \lambda_2= -\lambda_1, \, \lambda_3= 0\}.
\end{gather*}
Using $s_2$ (which is more general) we conclude that all systems with two parallel complex
straight lines and a real straight line as invariants curves have a Darboux invariant.
Moreover taking $\lambda_1=-i/2$ we obtain
\[
I_{16}(x,y,t)= e^{t} (x-i)^{i/2} (x+i)^{-i/2}.
\]
Using the polar form of the complex numbers it follows that
$(x-i)^{i/2} (x+i)^{-i/2}=e^{\arctan(1/x)}$ so the Darboux invariant is
$I_{16}(x,y,t)=e^{\arctan(1/x)+t}$.
In \cite{gasull} the authors already study the quadratic systems with $f=x^2+1=0$ as an
invariant curve, given by
$ \dot{x}=x^2+1$, $\dot{y}=Q(x,y)$,
with $Q$ an arbitrary polynomial of degree $2$. In this paper we have
$Q(x,y)=y(\alpha+\beta x+\gamma y)$. So the system studied here is a subcase of systems
(VI) in \cite{gasull}. In \cite{gasull} the study of those systems is divided in six
cases and since we have the invariant straight line $y=0$ there are seven possible phase
portraits. The case (VI.1) provides the phase portraits $1$ and $2$ of \cite[Fig. 1]{gasull},
i.e. the phase portraits CPL.1 and CPL.2 of Figure \ref{pp_6}; the case
(VI.2) gives the phase portrait $6$ of \cite[Fig. 1]{gasull}, i.e. the phase portrait
CPL.3 of Figure \ref{pp_6}; the case (VI.4) generates the phase portraits
$16$ and $17$ of \cite[Fig 1]{gasull}, i.e. the phase portraits CPL.4 and CPL.5 of
Figure \ref{pp_6}; the case (VI.5) gives the phase portrait $20$ of \cite[Fig. 1]{gasull},
i.e. the phase portrait CPL.6 of Figure \ref{pp_6}.
Finally the case (VI.6) provides the phase \cite[portrait 21 Fig. 1]{gasull}, i.e.
the phase portrait CPL.7 of Figure \ref{pp_6}.
\end{proof}
\begin{proposition} \label{(p)}
Each real planar quadratic differential system with two complex invariant straight lines that
intersects in a real point and a third invariant straight line having a Darboux can be written,
after an affine change of coordinates, as one of the following forms
\begin{itemize}
\item [(i)] {\rm (p.1)} with $\alpha_3(\beta-2\beta_3)\neq 0$ and Darboux invariant
\begin{equation*}
I_{17}(x,y,t)= e^{\alpha_3(\beta -2 \beta_3)\,t}
e^{-2 \gamma_3\arctan(y/x)}(x^2+y^2)^{\beta_3} y^{-\beta} .
\end{equation*}
\item [(ii)] {\rm (p.2)} with $c \neq 0$, $\alpha=-1$ and Darboux invariant
\begin{equation*}
I_{18}(x,y,t)= e^{-\arctan(y/x)-ct}
\end{equation*}
\end{itemize}
Moreover there are 5 non-equivalent phase portraits in the Poincar\'e disc for these systems.
They are described by p.1.1--p.1.3 and p.2.1, p.2.2 in Figure \ref{pp_6}.
\end{proposition}
\begin{proof}
Let $f_1=x+iy=0$ and $f_2=x-iy=0$ be the two complex straight lines that intersect at a
real point. We have two systems, (p.1), with $f_3=y$, and (p.2) with $f_3=y+a x+c$.
We shall do the calculations for (p.1), and for system (p.2) the computations are analogous.
we consider system (p.1) the cofactors of $f_1, f_2$ and $f_3$ are, respectively,
\begin{gather*}
k_1=(1/2)(\beta x+ 2 \gamma_3 y+2\alpha_3-i(\beta-2\beta_3)y),\\
k_2=(1/2)(\beta x+ 2 \gamma_3 y+2\alpha_3+i(\beta-2\beta_3)y),\\
k_3=\alpha_3+\beta_3 x+\gamma_3 y.
\end{gather*}
Solving equation \eqref{igualdade de darboux} the most general solution is
\begin{equation*}
\lambda_1=\beta_3+i \gamma_3, \quad \lambda_2=\beta_3-i \gamma_3, \quad
\lambda_3=-\beta, \quad s= \alpha_3(\beta-2\beta_3).
\end{equation*}
Hence assuming $\alpha_3(\beta-2\beta_3) \neq 0$ system (p.1) of Theorem \ref{teo1}
has the Darboux invariant
\begin{equation*}% \label{ip1}
I_{17}(x,y,t)=e^{\alpha_3 (\beta -2 \beta_3)\,t}y^{-\beta }
(x-i y)^{\beta_3-i \gamma_3} (x+i y)^{\beta_3+i \gamma_3}.
\end{equation*}
Using the polar form of the complex numbers it follows that
$$
(x-i y)^{\beta_3-i \gamma_3} (x+i y)^{\beta_3+i \gamma_3}
=e^{-2 \gamma_3\arctan(y/x)}\,(x^2+y^2)^{\beta_3}
$$
and we obtain the Darboux invariant
\[
I_{{17}}(x,y,t)= e^{\alpha_3(\beta -2 \beta_3)t}
e^{-2 \gamma_3\arctan(y/x)}\,(x^2+y^2)^{\beta_3} y^{-\beta}
\]
For system (p.2) the third invariant straight line is $f_3=y+a x+c$ with $c \neq 0$.
In this case the system has a Darboux invariant if and only if $\alpha=-1$, and with the
same reasoning applied above we obtain the invariant
\[
I_{18}(x,y,t)=e^{-\arctan(y/x)-c\,t}.
\]
We start the study of the global phase portraits with systems (p.1).
Since $\alpha_3 \neq 0$ we can take $\alpha_3 =1$. Systems (p.1) have at most two finite
singularities, namely $z_1=(0,0)$ and $z_2=(-2/\beta,0)$. When $\beta=0$ the point $z_2$ goes
to infinity.
The point $z_1$ is an unstable node and the eigenvalues associated to $z_2$ are $-1$ and
$(\beta-2\beta_3)/\beta$. So the point $z_2$ is either a stable node or a saddle.
In the local chart $U_2$ the origin is not a singularity for the compactfied system.
In the local chart $U_1$ the system compactified has only one infinity singularity
$u_1=(0,0)$ with eigenvalues $-\beta/2$ and $-(\beta-2\beta_3)/2$.
Then if $\beta(\beta-2\beta_3)>0$, $z_2$ is a saddle and $u_1$ is a stable node and the only
phase portrait is p.1.1 of Figure \ref{pp_6}, realizable for $\beta=1$, $\gamma_3=1$ and
$\beta_3=-1/2$. If $\beta(\beta-2\beta_3)<0$, $z_2$ is a stable node and $u_1$ is a saddle and the
corresponding phase portrait of this case is p.1.2 of Figure \ref{pp_6}, realizable for
$\beta=1$, $\gamma_3=1$ and $\beta_3=3/2$. Finally if $\beta=0$ then $z_2$ goes to the
infinity and $u_2$ becomes a semi hyperbolic saddle-node generating the phase portrait
p.1.3 of Figure \ref{pp_6}, which is realizable for $\beta=0$, $\gamma_3=1$ and $\beta_3=2$.
To study the global phase portraits of systems (p.2) we start with the infinity singular points.
In the local chart $U_1$ system (p.2) becomes
\begin{equation*}
\dot{u}=-c v(u^2+1) ,\quad
\dot{v}=-v (a \beta +a u+c u v+\beta c v+\beta u-1).
\end{equation*}
So the line $v = 0$ is filled up of singular points. The same happens in the local chart $U_2$.
In the finite part the point $(0,0)$ is the only singularity, with complex eigenvalues.
So the origin can be a node or a center. Both cases are described, respectively, by the
phase portraits p.2.1, realizable with $a=\beta=1$ and $c=2$, and p.2.2,
realizable with $a=1$, $\beta=0$ and $c=2$, of Figure \ref{pp_6}.
\end{proof}
Summarizing the nine propositions about quadratic systems with a invariant reducible cubic
and a Darboux invariant, we present table \ref{inshort}. In this table we expose the
relation between the normal forms and the phase portraits that can occur, as well as
the Figure where the corresponding phase portrait is given in this manuscript.
\begin{table}[http]
\caption{Table of relations among all the normal forms and the possible phase portraits of systems which have a Darboux invariant.}
\label{inshort}
\centering
\begin{tabular}{|c|l|l|}
\hline
Normal form & Cond. for a Darboux invariant & Possible phase portratis\\
\hline
(E.2) & $c=-1,\, \beta_1=2 \beta_2, \,\alpha_2 \neq 0$ & EL.2.1--EL.2.2 (Figure \ref{pp_1})\\
(H.2) & $c=-1, \, \gamma_1=2\gamma_2, \,\alpha_2 \neq 0$ & HL.2.1--HL.2.3 (Figure \ref{pp_1})\\
(H.3) & $c=0, \,\beta=-\gamma,\,A \alpha \neq 0$ & HL.3.1--HL.3.5 (Figure \ref{pp_1})\\
(H.3) & $\beta=\gamma=0,\,\alpha \neq 0$ & HL.3.6--HL.$3.9$ (Figure \ref{pp_1})\\
(H.4) & $A=2\beta,\,\alpha \neq 0$ & HL.2.1--HL.2.3 (Figure \ref{pp_1})\\
(P.1) & $\alpha_1-2\alpha_2 \neq 0$ & PL.1.1--PL.1.24 (Figure \ref{pp_2})\\
& & PL.1.25--PL.1.30 (Figure \ref{pp_3})\\
(P.2) & $\beta_1=\beta_2, \, \alpha_2=0, \,c \gamma_2 \neq 0$ & PL.2.1--PL.2.4 (Figure \ref{pp_3})\\
(P.2) & $c=\gamma_2=0, \,\alpha_2(\beta_2-\beta_2) \neq 0$ & PL.2.5--PL.$2.11$ (Figure \ref{pp_3})\\
(LV.1) & $\alpha (q-\beta) \neq 0$ & LVL.1.1--LVL.1.6 (Figure \ref{pp_4}) \\
(LV.2) & $c=-1$ & LVL.2.1--LVL.$2.17$ (Figure \ref{pp_4})\\
(LV.2) & $c=q=0, \, p \neq 0$ & DL.$1$--DL.$3$ (Figure \ref{pp_5}) \\
(LV.3) & $\alpha=-(\beta+1), \, c\beta \neq 0 $ & LVL.3.1 (Figure \ref{pp_4})\\
(RPL) & always has a Darboux invariant & RPL.$1$--RPL.$17$ (Figure \ref{pp_5})\\
(DL) & $\gamma=0, \, \alpha \neq 0$ & DL.$1$--DL.$3$ (Figure \ref{pp_5})\\
(CPL) & always has a Darboux invariant & CPL.$1$--CPL.$7$ (Figure \ref{pp_6})\\
(p.1) & $\alpha_3 (\beta-2\beta_3) \neq 0$ & p.1.1--p.1.3 (Figure \ref{pp_6})\\
(p.2) & $\alpha=-1, \, c\neq 0$ & p.2.1--p.2.2 (Figure \ref{pp_6})\\
\hline
\end{tabular}
\end{table}
By the end we prove Theorem \ref{theo2}. This result is about the differential systems
having an invariant cubic but that do not have a Darboux invariant.
\begin{proof}[Proof of Theorem \ref{theo2}]
First we consider systems of type (CE), i.e, the ones which has an invariant cubic of the
form $f=f_1f_2=0$ where $f_1=x^2+y^2+1$ and $f_2=a x+b y+c$.
By Theorem \ref{teo1} these systems can be written as
\begin{equation*}
\dot{x}=-(x^2+y^2+1)-2 \alpha_1 y (y+a x+c),\quad
\dot{y}=a (x^2+y^2+1)+2 \alpha_1 x (y+a x+c),
\end{equation*}
with $f_1=x^2+y^2+1$ and $f_2=y+a x+c$. The cofactors of $f_1$ and $f_2$ are
$k_1(x,y)=2(a y+x)$ and $k_2(x,y)=-2 \alpha_1 (a y-x)$, respectively.
So the cofactors have no constant terms, i.e., $k_1(0,0)=k_2(0,0)=0$.
The consequence of this is that equation \eqref{igualdade de darboux} has no solution
considering $s \neq 0$. Hence these systems do not have a Darboux invariant of the
form $e^{s t} f_1^{\lambda_1}f_2^{\lambda_2}$.
The proofs for the other systems are very similar.
In fact it suffices to observe that the cofactors of the invariant curves never have a
constant term.
\end{proof}
\subsection*{Acknowledgements}
J. Llibre was supported by the Ministerio de Ciencia, Innovaci\'on
y Universidades, Agencia Estatal de Investigaci\'on grant PID2019-104658GB-I00,
the Ag\`encia de Gesti\'o d'Ajuts Universitaris i de Recerca grant 2017SGR1617,
and the H2020 European Research Council grant MSCA-RISE-2017-777911.
R. Oliveira was supported by FAPESP “Projeto Tem\'atico" grant 2019/21181-0 and
by the CNPq grant 304766/2019-4 (Produtividade em Pesquisa).
We want thank to the referees for their comments and suggestions which help us
to improve the presentation of this article.
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