\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 82, pp. 1--37.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2008/82\hfil Quadratic systems]
{When singular points determine quadratic systems}
\author[J. C. Art\'{e}s, J. Llibre, N. Vulpe \hfil EJDE-2008/82\hfilneg]
{Joan C. Art\'{e}s, Jaume Llibre, Nicolae Vulpe} % in alphabetical order
\address{Joan C. Art\'{e}s \newline
Departament de Matem\`{a}tiques, Universitat Aut\`{o}noma de
Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain}
\email{artes@mat.uab.cat}
\address{Jaume Llibre \newline
Departament de Matem\`{a}tiques, Universitat Aut\`{o}noma de
Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain}
\email{jllibre@mat.uab.cat}
\address{Nicolae Vulpe \newline
Institute of Mathematics and Computer Science, Academy of Science
of Moldova, 5 Academiei str, Chi\c sin\u au, MD-2028, Moldova}
\email{nvulpe@mail.md}
\thanks{Submitted September 10, 2006. Published May 30, 2008.}
\thanks{The first two authors are supported by
grants MTM2005-06098-C02-01 from \hfill\break\indent
MEC/FEDER, and 2005SGR-00550 from CICYT.
The third author is supported \hfill\break\indent
by grant CERIM-1006-06 from CRDF-MRDA}
\subjclass[2000]{34C05, 34C08}
\keywords{Quadratic systems; singular points}
\begin{abstract}
When one considers a quadratic differential system, one realizes
that it depends on 12 parameters of which one can be fixed by
means of a time change. One also can notice that fixing 4 finite
real singular points plus 3 infinite real ones (all its possible
singular points) implies to fix 11 conditions, that is, 11
equations that the parameters must satisfy. Since these conditions
are linear with respect to the parameters, it is obvious to think
that the system will be determined, except that the fixed conditions
are incompatible with a quadratic differential system having
finitely many singular points.
In this paper we prove exactly this. That is, if we fix the
position of the 7 singular points of a quadratic differential system
in a distribution that does not force an infinite number of finite
singular points, then the system is completely determined, and
consequently its phase portrait is also determined. This determination
includes the local behavior of all singular points, even if they are weak
focus or centers, the global behavior of separatrices, and even
the existence or not of limit cycles. This also implies that limit
cycles are sensitive to small perturbations of the coordinates of
singular points, even if they are far from the singular points.
The result of the paper goes far beyond this, since we state that
this result is independent of the fact that the fixed singular points are
real or complex, and it does not mind if the infinite singular points
are simple or multiple due to the collision of several infinite singular
points. Only when some data is lost due to the collision of finite
singular points or to the collision of some finite singular points with
infinite ones, this adds free parameters to the set of parameters at
the same rate than the number of finite singular points are
lost.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction and statement of main results}
We consider the real polynomial differential systems
\begin{equation}\label{s:gen}
\frac {dx}{dt}= P(x,y),\quad
\frac {dy}{dt}= Q(x,y),
\end{equation}
where $P$ and $Q$ are polynomials in $x$ and $y$ with real
coefficients; i.e. $P,Q\in \mathbb{R}[x,y]$. We say that systems
\eqref{s:gen} are {\it quadratic} if
$\max\big(\deg(P),\deg(Q)\big)=2$.
Quadratic differential systems have been studied from many
different points of view (the following lists clearly are not
complete): studying their finite singular points \cite{Berl,
Berl2, Blt_Vlp1, Blt_Vlp3, Curtz1, Curtz2, Curtz3, Vlp:1991,
Art_Llib_Vlp}, studying their infinite singular points
\cite{Koj_Rn, Nik_Vlp, Dana_Vlp1}, studying systems with limit
cycles \cite{Artes-Cherkas-Llibre, CFLL}, studying systems with
invariant straight lines \cite{Popa_Sib1, Popa_Sib2, Popa_Sib3,
Dana_Vlp2, Dana_Vlp5}, studying systems with centers \cite{Dulac,
Kapteyn1, Frommer, Saharn, Bul_Vulp_Sib, Vlp:1991, Schlom, Zol},
and systems with weak focus \cite{AL, Llibre-Schlomiuk:2004, ALS},
studying systems with invariant algebraic curves or first
integrals \cite{Art_Lb1, LL}, classifying phase portraits
according to the number of finite singular points \cite{GSL,
Reyn1, Reyn2, Reyn3}, classifying phase portraits according to the
structural stability of the portrait \cite{AKL}, and many others
ways up to more than 1000 papers have been published on these
systems \cite{Reyn}.
But up to now no one seems to have noticed the relation between
the number of parameters of a quadratic differential systems, and
the number of conditions that are fixed by determining the
situation of the 4 finite singular points and 3 infinite ones that
a quadratic system can have. Intuitively one easily realizes that
fixing 11 conditions forces a linear system of equations which if
it is not incompatible, it depends just on one parameter and it
determines uniquely a phase portrait since one parameter can be
removed by means of a time change.
This case could be proved by means of simple algebra tools, and so
we tried to go a bit further, and realized that it was not
important whether the singular points were real or complex. The
conditions would be the same. Moreover it did not matter whether
the infinite singular points where simple or multiple, meanwhile
their multiplicity was due only on the collision of infinite
singular points.
It was also clear that when a finite singular point collided with
another finite singular point or infinite, we would lose two data (of the
coordinates of the singular point) but win one data from the
non-hyperbolicity. Thus the system should now have one free parameter. However the conditions are no more linear and thus it is
not so obvious that the result should be that. When one adds more
collisions between singular points, and specially if several
points collide all together, it is even less obvious how many
degrees of freedom will be obtained.
In order to compute this we have used the theory of invariants
developed by Sibirsky and his disciples (cf. \cite{Sib1},
\cite{Vlp1}, \cite{Popa5}, \cite{Baltag}, \cite{Calin}) and completed
for the quadratic differential systems by Schlomiuk and Vulpe
\cite{Dana_Vlp1} when dealing with infinite singular points and by
Artes, Llibre and Vulpe \cite{Art_Llib_Vlp} when dealing with
finite singular points.
The main result of this paper is the following.
\begin{theorem}\label{th:Main}
We consider a family of quadratic
systems \eqref{s:gen} depending on $12$ parameters. Assume that
one parameter is removed via a time rescaling and that all the
coordinates of the singularities (finite and infinite, real and/or
complex, simple and/or multiple) are fixed and that this does not force
an infinite number of finite singular points. Then we get a family
of quadratic systems whose number of free parameters is four minus
the number of distinct finite singular points (real or complex).
\end{theorem}
In Section 2 we introduce some preliminary results and definitions
needed for the rest of the work, and which deal mainly with the
theory of invariants. The proof of the Main Theorem is split in five
sections from the 3 to the 7 since we develop separately the cases with
four finite singular points (real or complex) in Section 3, three
finite singular points (real or complex) in Section 4, two finite
singular points (real or complex) in Section 5, less than two finite
singular points in Section 6, and systems with
the infinite full of singular points which have been skipped into the
previous sections are considered all together in Section 7.
It is worth to note that a quadratic system with 4 distinct finite
singular points (real or complex) has its phase portrait
completely determined once these four points and the infinite
points (real or complex, simple or multiple) are fixed. That is,
whether the singular points are saddles, nodes, foci or centers is imbedded
by the position of the singular points. Even more, whether a focus
is strong or weak, whether there is a separatrix connection or
not, an invariant straight line, and even the existence of limit
cycles comes determined in this case only by the position of the
singular points.
This implies that the perturbation of one singular point
(finite or infinite, real or complex) may affect the phase
portrait. It may, for example, imply the born or death of limit
cycles.
To illustrate this last case, we are going to provide an example.
In \cite{Artes-Cherkas-Llibre} it is proved that the system:
\begin{equation}\label{sys:cherkas}
\begin{gathered}
\dot x=1+xy, \\
\dot y=0.722 + 15.28x + 8.4y - 12x^2 - 1.398xy + 3y^2,
\end{gathered}
\end{equation}
has exactly three concentric limit cycles of visible size around the
singular point $p_1=(1,-1)$. The other finite singular point is
$p_2=(-0.7571298123634432\ldots,\\ 1.320777472595376\ldots)$
and there are two finite complex singular points being\\
$p_{3,4}=(0.515231572848387\ldots \pm
0.2544224724528470\ldots i, \\
-1.56038873629768\ldots$ $ \pm
0.770523355317071\ldots i)$. The infinite singular points are the
$(0,0)$ of the local chart $U_2$ and the points $z_1=(-2.1247979307270,0)$
and $z_2=(2.8237979307270,0)$ of the local chart $U_1$.
We must remark that the study of the limit cycles of polynomial differential
systems (even for quadratic) is still far from be completed, and that
numerical tools can only provide evidence of existence of large
limit cycles but are useless to detect infinitesimal ones. So we
will limit now to talk about these large limit cycles which can be
observed in \eqref{sys:cherkas}.
Now we will perturb a little one of the infinite singular points.
Concretely we will take $z=z_2+(\varepsilon,0)$. With the help
of the numerical program P4 \cite{P4}, we can
numerically show that when $\varepsilon=0.00018$ the three large limit
cycles persist, at $\varepsilon=0.0002$ only one large
limit cycle persists, and at $\varepsilon=0.001$ there is no large limit
cycles. Moreover the inner limit cycle of \eqref{sys:cherkas} has
disappeared in a Hopf bifurcation changing the stability of the
focus $p_1$ and the central and outer limit cycles have collapsed
in a double semi--stable limit cycle.
It is also remarkable that all known situations of quadratic
differential systems having the maximum number of known limit
cycles correspond to cases with four finite singular points (two
of them complex). That is they correspond to cases that are
completely determined (up to a time change) by the location of the
singular points. It is also known that some conditions (like
having an invariant straight line) restrict the number of possible
limit cycles (in the case of one invariant straight line the
maximum is one \cite{Cop, CL}, and with two invariant straight
lines none is possible \cite{Cop66}). It may happen that the
maximum number of limit cycles is also related with the number of
parameters which are not determined by the position of the singular
points of the system.
\section{Some preliminary results and definitions}\label{Preliminary}
\subsection{Zero-cycles associated to finite and infinite singularities}
Consider real quadratic systems of the form
\begin{equation}\label{sys:gen}
\begin{gathered}
\frac {dx}{dt}=p_0+ p_1(x,y)+\,p_2(x,y)\equiv P(x,y), \\
\frac {dy}{dt} =q_0+ q_1(x,y)+\,q_2(x,y)\equiv Q(x,y),
\end{gathered}
\end{equation}
with homogeneous polynomials $p_i$ and $q_i$ $(i=0,1,2)$ of degree $i$ in
$x$ and $y$, where
\begin{gather*}
p_0=a_{00},\quad p_1(x,y)= a_{10}x+ a_{01}y,\quad p_2(x,y)= a_{20}x^2 +2
a_{11}xy + a_{02}y^2,\\
q_0=b_{00},\quad q_1(x,y)= b_{10}x+ b_{01}y,\quad q_2(x,y)= b_{20}x^2 +2
b_{11}xy + b_{02}y^2.\\
\end{gather*}
We associate with the two polynomials $P, Q\in \mathbb R[x,y]$
defining systems \eqref{sys:gen}, the homogeneous polynomials
$P^*, Q^*$ in $X, Y, Z$ of
degree $2$ with real coefficients defined as follows
$$
P^*(X,Y,Z) = Z^2P(X/Z,Y/Z),\quad Q^*(X,Y,Z) = Z^2Q(X/Z,Y/Z),
$$
and denote $C^*(X,Y,Z)=YP^*(X,Y,Z)-X Q^*(X,Y,Z)$.
We shall use the notions of \textit{zero-cycle} and
\textit{divisor} in order to describe the number and multiplicity
of singularities of a quadratic system \eqref{sys:gen} (for the
definitions of these notions see \cite{Pal-Schlomiuk:2001a}). The
notions of zero-cycle and divisor were used for classification
purposes of planar quadratic differential systems by Pal and
Schlomiuk \cite{Pal-Schlomiuk:2001a}, Llibre and Schlomiuk~\cite
{Llibre-Schlomiuk:2004}, Schlomiuk and Vulpe \cite{Dana_Vlp1} and
by Artes and Llibre and Schlomiuk~\cite{ALS}.
Following \cite{Pal-Schlomiuk:2001a} (see also \cite{Dana_Vlp1})
we define here the next zero-cycle and divisor.
For a system $(S)$ belonging to family \eqref{sys:gen} we denote
$\sigma(P,Q)= \{w\in\mathbb{C}_2\ |\ P(w)= Q(w)=0\}$ and we define the
zero-cycle $ \mathcal{D}^f_{{}_S}(P,Q) =
\sum_{w\in\sigma(P,Q)}I_w(P,Q)w, $\ where $I_w(P,Q)$ is the
intersection number or multiplicity of intersection at $w$. It is
clear that for a non--degenerate quadratic system
$\deg(\mathcal{D}^f_{{}_S})=\sum I_w(P,Q)\le4$. For a degenerate
system (i.e. $\gcd(P,Q)\ne \mathrm{constant}$) the zero-cycle
$\mathcal{D}^f_{{}_S}(P,Q)$ is undefined.
Assume that a system $(S)$ is such that $P(x,y)$ and $Q(x,y)$ are
relatively prime over $\mathbb C$ and that $yp_2- xq_2$ is not
identically zero. The following divisor on the line at infinity $Z=0$
is then well defined
$$
\mathcal{D}^\infty_{{}_S}
= \sum_{w\in\{ Z = 0\}} \begin{pmatrix}
I_w(P^*,Q^*)\\
I_w(C^*,Z)\end{pmatrix}w.
$$
We note that the zero-cycle $ \mathcal{D}^f_{{}_S}$ describes
the number of finite singularities which could arise from a
perturbation of $(S)$ from singularities in the phase plane of
$(S)$. On the other hand the divisor $\mathcal{D}^\infty_{{}_S}$
describes the number of singularities which could arise from a
perturbation of $(S)$ from singularities at infinity of $(S)$ in
both the finite plane and at infinity.
\subsection{Affine invariant polynomials associated with
infinite singularities}
It is known that on the set $QS$ of all
quadratic differential systems \eqref{sys:gen} acts the group
$Af\,f(2,\mathbb{R})$ of the affine transformation on the plane \mbox{(cf.
\cite {Dana_Vlp1})}. For every subgroup $G\subseteq Aff(2,\mathbb{R})$ we
have an induced action of $G$ on $QS$. We can identify the set
$QS$ of systems \eqref{sys:gen} with a subset of $\mathbb{R}^{12}$ via
the map $QS\to \mathbb{R}^{12}$ which associates to each
system \eqref{sys:gen} the 12-tuple $(a_{00},\ldots,b_{02})$ of
its coefficients.
For the definitions of a $GL$-comitant and invariant as well as
for the definitions of a $T$-comitant and a $CT$-comitant we refer
the reader to the paper \cite {Dana_Vlp1}. Here we shall only
construct the necessary $T$-comitants and $CT$-comitants
associated to configurations of finite and infinite singularities
(including multiplicities) of quadratic systems \eqref{sys:gen}.
Using the so called {\it transvectant of order $k$} (see {\rm
\cite{Gr_Yng}, \cite{Olver}}) of two polynomials
$f,g\in \mathbb{R}[a,x,y]$,
$$
(f,g)^{(k)}=
\sum_{h=0}^k (-1)^h {k\choose h}
\frac{\partial^k f}{\partial x^{k-h}\partial y^h}\
\frac{\partial^k g}{\partial x^h\partial y^{k-h}},
$$
we shall construct the following invariant polynomials
\begin{align*}
{C}_i(a,x,y)&= y p_i(x,y)-x q_i(x,y),\ i=0,1,2;\\
K(a,x,y)&= \mathop{\rm Jacob}\big(p_2(x,y),q_2(x,y)\big)/4;\\
\mu_0(a)&= \mathop{\rm Res}{}_x(p_2,q_2)/y^4 = \mathop{\rm Discrim}\big(K(a,x,y)\big)/16;\\
M(a,x,y)&= 2\,\mathop{\rm Hess}\big(C_2(a,x,y)\big);\\
\eta(a)&= \mathop{\rm Discrim}\big(C_2(a,x,y)\big);\\
H(a,x,y)&= -\mathop{\rm Discrim}(\alpha p_2(x,y)+\beta q_2(x,y))
\big|_{\{\alpha=y,\, \beta=-x\}};\\
\kappa(a)&= (M,K)^{(2)}; \\
\kappa_1(a)&= (M,C_1)^{(2)};\\
L(a,x,y)&= 16K+8H-M;\\
K_1(a,x,y)&= p_1(x,y)q_2(x,y)-p_2(x,y)q_1(x,y).
\end{align*}
Consider the differential operator
$\mathcal{L}= x\cdot \mathbf{L}_2 -y\cdot\mathbf{L}_1$ acting
on $\mathbb R[a,x,y]$ constructed in \cite{Blt_Vlp2}, where
\begin{gather*}
\mathbf{L}_1= 2a_{00}\frac{\partial}{\partial a_{10}} +
a_{10}\frac{\partial}{\partial a_{20}} +
\frac{1}{2}a_{01}\frac{\partial}{\partial a_{11}} +
2b_{00}\frac{\partial}{\partial b_{10}} +
b_{10}\frac{\partial}{\partial b_{20}} +
\frac{1}{2}b_{01}\frac{\partial}{\partial b_{11}},\\
\mathbf{L}_2= 2a_{00}\frac{\partial}{\partial a_{01}} +
a_{01}\frac{\partial}{\partial a_{02}} +
\frac{1}{2}a_{10}\frac{\partial}{\partial a_{11}} +
2b_{00}\frac{\partial}{\partial b_{01}} +
b_{01}\frac{\partial}{\partial b_{02}} +
\frac{1}{2}b_{10}\frac{\partial}{\partial b_{11}}.
\end{gather*}
Using this operator we construct the following polynomials
\[
\mu_i(a,x,y) =\frac{1}{i!} \mathcal{L}^{(i)}(\mu_0), \ i=1,\dots,4,\quad
\text{where}\quad \mathcal{L}^{(i)}(\mu_0)=\mathcal{L}(\mathcal{L}^{(i-1)}(\mu_0)).
\]
These polynomials are in fact comitants of
systems \eqref{sys:gen} with respect to the group $GL(2,\mathbb
R)$ (see \cite{Blt_Vlp2}). Their geometrical meaning is revealed
in Lemmas \ref{lem:mu_i-Infty} and \ref{lem:mu_i-(0,0)} below.
\subsection{Some useful assertions}
\begin{lemma}[\cite{Blt_Vlp2,Dana_Vlp1}] \label{lem:mu_i-Infty}
The system $P^*(X,Y,Z)=Q^*(X,Y,Z)=0$ possesses $m$ ($1\le m \le4$) solutions
$[X_i:Y_i:Z_i]$ with $Z_i=0$ $(i=1,\ldots,m$) (considered with multiplicities)
if and only if for every $ i\in\left\{0,1,\ldots,m-1\right\}$ we have
$\mu_i(a,x,y)=0$ in $\mathbb R[a,x,y]$ and $\mu_m(a,x,y)\ne0$.
\end{lemma}
\begin{lemma}[\cite{Blt_Vlp2}] \label{lem:mu_i-(0,0)}
The point $M_0(0,0)$ is a singular point of multiplicity $k$
($1\le k \le4$) for a quadratic system \eqref{sys:gen} if and
only if for every $ i\in\left\{0,1,\ldots,k-1\right\}$ we have
$\mu_{4-i}(a,x,y)=0$ in $\mathbb R[a,x,y]$ and
$\mu_{4-k}(a,x,y)\ne0$.
\end{lemma}
\begin{remark} \label{rem:2FtoInf}
{\rm We note that according to Lemma \ref{lem:mu_i-Infty} at least
two finite singular points of a quadratic system have gone to
infinity if and only if $\mu_0=\mu_1=0$.}
\end{remark}
\begin{remark} \label{rem:lin-fct-p2,q2}
{\rm Assume that the polynomials $p_2(x,y)$ and $q_2(x,y)$ of
systems \eqref{sys:gen} have a common linear factor $\alpha
x+\beta y$. Then these systems have the infinite singular point
$N(-\beta,\alpha,0)$ and via a rotation we can assume that this
infinite singularity is located in the direction $x=0$, i.e. $x$
will be a factor of $p_2(x,y)$ and $q_2(x,y)$. This implies
$a_{02}=b_{02}=0$ for systems \eqref{sys:gen}.}
\end{remark}
The next lemma follows directly from \cite[Theorem 5.1]{Dana_Vlp1}.
\begin{lemma} \label{lem:inf-config}
Assume that at least two finite singular points of a quadratic
system with finite number of infinite singularities have gone to
infinity (i.e. $\mu_0=\mu_1=0$). Then the configurations of the
infinite singularities are given by the following conditions.
\begin{itemize}
\item[(a)] 3 real distinct points
\begin{align*}
& {2\choose 1}p+{0\choose 1}q+{0\choose 1}r
\Leftrightarrow \mu_2\ne0,\, \kappa\ne0;\\
&{1\choose 1}p+{1\choose 1}q+{0\choose 1}r \Leftrightarrow
\mu_2\ne0,\ \kappa=0;\\
&{3\choose 1}p+{0\choose 1}q+{0\choose 1}r \Leftrightarrow
\mu_2=0,\, \mu_3\ne0,\, \kappa\ne0;\\
&{2\choose 1}p+{1\choose 1}q+{0\choose 1}r \Leftrightarrow
\mu_2=0,\, \mu_3\ne0,\, \kappa=0;\\
&{4\choose 1}p+{0\choose 1}q+{0\choose 1}r \Leftrightarrow
\mu_2=\mu_3=0,\, \mu_4\ne0,\ \kappa\ne0;\\
&{3\choose 1}p+{1\choose 1}q+{0\choose 1}r \Leftrightarrow
\mu_2=\mu_3=0,\, \mu_4\ne0,\ \kappa=0,\ K_1\ne0;\\
&{2\choose 1}p+{2\choose 1}q+{0\choose 1}r \Leftrightarrow
\mu_2=\mu_3=0,\, \mu_4\ne0,\, \kappa=0,\, K_1=0;
\end{align*}
\item[(b)] one real and 2 complex singular points
\begin{align*}
& {2\choose 1}p+{0\choose 1}q^c+{0\choose 1}r^c
\Leftrightarrow \mu_2\ne0,\, \kappa\ne0;\\
&{0\choose 1}p+{1\choose 1}q^c+{1\choose 1}r^c \Leftrightarrow
\mu_2\ne0,\, \kappa=0;\\
&{3\choose 1}p+{0\choose 1}q^c+{0\choose 1}r^c \Leftrightarrow
\mu_2=0,\, \mu_3\ne0;\\
&{4\choose 1}p+{0\choose 1}q^c+{0\choose 1}r^c \Leftrightarrow
\mu_2=\mu_3=0,\, \mu_4\ne0,\, \kappa\ne0;\\
&{0\choose 1}p+{2\choose 1}q^c+{2\choose 1}r^c \Leftrightarrow
\mu_2=\mu_3=0,\, \mu_4\ne0,\, \kappa=0;
\end{align*}
\item[(c)] one double and one simple real singular points
\begin{align*}
& {2\choose 1}p+{0\choose 2}q \Leftrightarrow
\mu_2\ne0,\, \kappa\ne0;\\
&{0\choose 1}p+{2\choose 2}q \Leftrightarrow
\mu_2\ne0,\, \kappa=0,\, L\ne0;\\
&{1\choose 1}p+{1\choose 2}q \Leftrightarrow
\mu_2\ne0,\, \kappa=0,\, L=0;\\
&{3\choose 1}p+{0\choose 2}q \Leftrightarrow
\mu_2=0,\, \mu_3\ne0,\, \kappa\ne0;\\
&{0\choose 1}p+{3\choose 2}q \Leftrightarrow
\mu_2=0,\, \mu_3\ne0,\ \kappa=0,\, L\ne0;\\
&{2\choose 1}p+{1\choose 2}q \Leftrightarrow
\mu_2=0,\, \mu_3\ne0,\ \kappa=0,\, L=0,\, \kappa_1\ne0;\\
&{1\choose 1}p+{2\choose 2}q \Leftrightarrow
\mu_2=0,\, \mu_3\ne0,\ \kappa=0,\, L=0,\, \kappa_1=0;\\
&{4\choose 1}p+{0\choose 2}q \Leftrightarrow
\mu_2=\mu_3=0,\, \mu_4\ne0,\, \kappa\ne0;\\
&{0\choose 1}p+{4\choose 2}q \Leftrightarrow
\mu_2=\mu_3=0,\, \mu_4\ne0,\, \kappa=0,\, L\ne0;\\
&{3\choose 1}p+{1\choose 2}q \Leftrightarrow
\mu_2=\mu_3=0,\, \mu_4\ne0,\, \kappa=0,\, L=0,\, \kappa_1\ne0;\\
&{1\choose 1}p+{3\choose 2}q \Leftrightarrow
\mu_2=\mu_3=0,\, \mu_4\ne0,\, \kappa=0,\ L=0,\, \kappa_1=0,\, K_1\ne0;\\
&{2\choose 1}p+{2\choose 2}q \Leftrightarrow
\mu_2=\mu_3=0,\, \mu_4\ne0,\, \kappa=0,\, L=0,\, \kappa_1=0,\, K_1=0;
\end{align*}
\item[(d)] one real triple point:
\begin{gather*}
{2\choose 3}p \Leftrightarrow \mu_2\ne0;\quad
{3\choose 3}p \Leftrightarrow \mu_2=0,\\
\mu_3\ne0; \quad {4\choose 3}p \Leftrightarrow
\mu_2=\mu_3=0,\, \mu_4\ne0.
\end{gather*}
\end{itemize}
\end{lemma}
\begin{lemma}\label{lem:C2-ISPs}
The polynomial $C_2(x,y)=yp_2(x,y)-xq_2(x,y)\not\equiv0$ is completely
determined up to a time rescaling by the coordinates of the three infinite
singular points (real and/or complex, simple or multiple).
\end{lemma}
\begin{proof}
It is known that the coordinates of infinite singular points of a
quadratic system \eqref{sys:gen} are given by the linear factors over $\mathbb{C}$ of
the polynomial
\begin{equation}\label{factoriz:C2}
\begin{aligned}
C_2(x,y)&=yp_2(x,y)-xq_2(x,y)\\
&=(u_1x+v_1y)(u_2x+v_2y)(u_3x+v_3y)\\
&=Ux^3+U_vx^2y+V_uxy^2 +Vy^3,
\end{aligned}
\end{equation}
where $U=u_1u_2u_3$, $U_v=u_1u_2v_3+u_1v_2u_3+v_1u_2u_3$,
$V=v_1v_2v_3$, $V_u=u_1v_2v_3+v_1u_2v_3+v_1v_2u_3$.
Since $C_2(x,y)\not\equiv0$ we have $U^2+U_v^2+V_u^2+V^2\ne0$.
If $U\ne0$ we may set $U=\theta$ and take $u_1=\theta$,
$u_2=u_3=1$. Then the infinite singular points (real or complex)
$N_i( k_i,1,0)$ $(i=1,2,3)$ completely determine $v_1=-\theta
k_1$, $v_2=- k_2$ and $v_3=- k_3$. Thus $U=\theta$,
$U_v=-\theta(k_1+k_2+k_3)$, $V_u= \theta(k_1K_2+k_1k_3+k_2k_3)$,
$V=-\theta k_1k_2k_3$.
If $U=0\ne U_v$ (i.e. $N_1(0,1,0)$ is an infinite singular point of systems
\eqref{sys:gen}) we may set $U_v=\theta$ and take $u_1=0$, $v_1=u_3=1$,
$u_2=\theta$. Then the remaining infinite singular points (real or complex)
determine $v_2=-\theta k_2$ and $v_3=-k_3$. As in the previous case
(and also the next two) the coefficients of $C_2$ become completely determined
(up to a multiplication of the parameter $\theta$) by the infinite singularities.
If $U=U_v=0\ne V_u$ (i.e. $N_1(0,1,0)$ is at least a double infinite singular
point of systems \eqref{sys:gen}), we may set $V_u=\theta$ and take
$u_1=u_2=0$, $v_1=v_2=1$, $u_3=\theta$ and the remaining infinite singular
point (which must be real) determines $v_3=-\theta k_3$.
If $U=U_v=V_u=0$ (then $V\ne0$) we may set $V=\theta$ and the only infinite
singularity is the point $N_1(0,1,0)$, which is at least of multiplicity three.
\end{proof}
\begin{remark} \label{rem:admissible-trans}
{\rm We are trying to determine
the number of free parameters of quadratic
systems once the coordinates of the singular points and their
configuration is fixed. In order to simplify calculations we
shall use the group $Af\,f(2,\mathbb{R})$ of affine transformations. We say
that an affine transformation is {\it admissible} if it is defined
using only the coordinates of some singular points (finite or
infinite). It is clear that an admissible affine transformation
keeps the number of free parameters of the respective family of systems.}
\end{remark}
\section{Quadratic systems with four distinct finite singularities}
\subsection{Systems with four real simple singular points}
Evidently a quadratic system with four real simple singular points
can be brought via an admissible (in the sense of Remark \ref{rem:admissible-trans})
affine transformation to the form
\begin{equation}\label{CS:r1r1r1r1}
\begin{gathered}
\dot x=cx+d y - cx^2 +2 h xy - dy^2\equiv P(x,y),\\
\dot y=ex+f y - ex^2 +2 mxy - fy^2\equiv Q(x,y).
\end{gathered}
\end{equation}
Each system from this family has the singular points $M_1(0,0)$, $M_2(1,0)$,
$M_3(0,1)$ and $M_4(\alpha,\beta)$. Now will find the dependence among the coefficients of systems \eqref{CS:r1r1r1r1} and the parameters $\alpha$ and $\beta$ of the fourth singular point. Since $\alpha\beta\ne0$ (we cannot have three distinct singular points placed on one line) from the identities
$P(\alpha,\beta)=Q(\alpha,\beta)=0$ we obtain
$$
h=\frac{c\alpha(\alpha-1)+d\beta(\beta-1)}{2\alpha\beta},\quad
m=\frac{e\alpha(\alpha-1)+f\beta(\beta-1)}{2\alpha\beta}.
$$
Therefore after a time rescaling ($t\to \alpha\beta t_1$) and some
re--parametrization ($c\alpha\to c$,\ $e\alpha\to e$,\ $d\beta\to d$,\
$f\beta\to f$) we get the following family of systems
\begin{equation}\label{CS:r1r1r1r1-a}
\begin{gathered}
\dot x=c\,\beta x(1-x)+d\,\alpha y(1-y)+\big[c(\alpha-1)+d(\beta-1)\big]xy,\\
\dot y=e\,\beta x(1-x)+f\alpha y(1-y)+\big[e(\alpha-1)+ f(\beta-1)\big]xy.
\end{gathered}
\end{equation}
Evidently each system of this family possesses the singular points
$M_1(0,0)$,
$M_2(1,0)$, $M_3(0,1)$ and $M_4(\alpha,\beta)$ and for this family by Lemma
\ref{lem:mu_i-Infty} the following condition must be satisfied
\begin{equation}\label{cond:mu_0}
\mu_0=\alpha\beta(\alpha+\beta-1)(cf-de)^2\ne0,
\end{equation}
otherwise the systems become degenerate.
Since for systems \eqref{CS:r1r1r1r1-a} we have
$$
C_2(x,y)=e\beta x^3+[e(1-\alpha)+f(1-\beta)-c\beta]x^2y+
[c(\alpha-1)+d(\beta-1)+f\alpha]xy^2-d\alpha y^3,
$$
considering the factorization \eqref{factoriz:C2} we get the following system
of linear (with respect to the parameters $c,d,e$ and $f$) equations
\begin{gather*}
e\,\beta=U,\quad e(1-\alpha)+f(1-\beta)-c\beta=U_v,\\
d\,\alpha=-V,\quad c(\alpha-1)+d(\beta-1)+f\alpha=V_u.
\end{gather*}
Solving this system we obtain
\begin{gather*}
c=-\frac{\alpha^2(\alpha-1)U+ \alpha^2\beta
U_v+\alpha\beta(\beta-1)V_u+\beta(\beta-1)^2V}
{\alpha\beta(\alpha+\beta-1)},\quad d=-\frac{V}{\alpha},\\
f=\frac{\alpha(\alpha-1)^2U+ \alpha\beta(\alpha-1)
U_v+\alpha\beta^2)V_u+\beta^2(\beta-1)V} {\alpha\beta(\alpha+\beta-1)},\quad
e=\frac{U}{\beta},
\end{gather*}
where $\alpha\beta(\alpha+\beta-1)\ne0$ by condition \eqref{cond:mu_0}.
Thus taking into account Lemma \ref{lem:C2-ISPs} we obtain that the
coefficients of systems \eqref{CS:r1r1r1r1-a} depend exclusively on the
coordinates of their singular points (finite and infinite).
\subsection{Systems with two real simple and two complex singular
points}\label{Subsec:1+2c}
By \cite{Art_Llib_Vlp} such a
quadratic system can be brought via an affine transformation to
the form
\begin{equation}\label{CS:r1r1c1c1a}
\begin{gathered}
\dot x= a -(a+g)x + gx^2 +2 h xy + ay^2,\\
\dot y= b -(b+l)x + lx^2 +2 m xy +b y^2.
\end{gathered}
\end{equation}
These systems possess the singular points $M_{1,2}(0,\pm i)$,
$M_3(1,0)$ and the fourth one, whose coordinates we will force to
be $(\alpha,\beta)$.
First we observe that $\alpha\ne0$, otherwise from the relation
$P(0,\beta)=Q(0,\beta)=0$ we obtain $a(1+\beta^2)=b(1+\beta^2)=0$ and this
leads to degenerate systems. So $\alpha\ne0$ and we shall consider two
subcases $\beta\ne0$ and $\beta=0$.
\subsubsection{The case $\beta\ne0$}
From the identities $P(\alpha,\beta)=Q(\alpha,\beta)=0$ for
systems \eqref{CS:r1r1c1c1a} we obtain
$$
h=\frac{(\alpha-1)(a-g\alpha)-a\beta^2}{2\alpha\beta},\quad
m=\frac{(\alpha-1)(b-l\alpha)-b\beta^2}{2\alpha\beta}.
$$
Therefore we get the following family of systems
\begin{equation}\label{CS:r1r1c1c1-a}
\begin{gathered}
\dot x=a(1-x+y^2)+gx(x-1)+\frac{1}{\alpha\beta}
\big[(\alpha-1)(a-g\alpha)-a\beta^2\big]xy,\\
\dot y=b(1-x+y^2)+\ lx(x-1)+
\frac{1}{\alpha\beta}\big[(\alpha-1)(b-l\alpha)-b\beta^2\big]xy.
\end{gathered}
\end{equation}
Evidently each system of this family possesses the singular points
$M_{1,2}(0,\pm i)$, $M_3(1,0)$ and $M_4(\alpha,\beta)$ and for this family by
Lemma \ref{lem:mu_i-Infty} the following condition must be satisfied
$\mu_0=\big[(\alpha-1)^2+\beta^2\big](al-bg)^2/(\alpha\beta^2)\ne0$.
We shall consider now the infinite singular points.
For systems \eqref{CS:r1r1c1c1-a} we have
\begin{align*}
C_2(x,y)&=-l
x^3+\frac{1}{\alpha\beta}\big[(\alpha-1)(l\alpha-b)
+\beta(g\alpha+b\beta)\big]x^2y\\
&\quad +\frac{1}{\alpha\beta}\big[(\alpha-1)(a-g\alpha)
-\beta(b\alpha+a\beta)\big]xy^2+ay^3.
\end{align*}
Using the factorization \eqref{factoriz:C2} and solving the respective
linear system with respect to the parameters $a,b,g$ and $l$ we obtain
\begin{gather*}
g=\frac{ (1-\alpha+\beta^2)^2 V+\alpha^2\beta(\alpha-1)U
+\alpha\beta^2(U_v\alpha+V_u\beta)-\alpha\beta(\alpha-1)V_u}
{\alpha\big[(\alpha-1)^2+\beta^2\big]},\quad a= V,\\
b=-\frac{ \beta(1-\alpha+\beta^2) V+\alpha(\alpha-1)^2U
+\alpha\beta^2V_u+\alpha\beta(\alpha-1)U_v}
{(\alpha-1)^2+\beta^2},\quad \quad l=-U.
\end{gather*}
Therefore considering Lemma \ref{lem:C2-ISPs} we obtain the family
of systems which depend exclusively on the coordinates of their
singular points (finite and infinite).
\subsubsection{The case $\beta=0$}
In this case from the identities $P(\alpha,0)=Q(\alpha,0)=0$ for systems
\eqref{CS:r1r1c1c1a} we obtain
$$
(1-\alpha)(a-g\alpha)=0,\quad (1-\alpha) (b-l\alpha)=0,
$$
and since $1-\alpha\ne0$ (otherwise the fourth point coincides with $M_3(1,0)$)
we get the following family of systems
\begin{gather*}
\dot x=g(x-1)(x-\alpha)+2hxy +g\alpha y^2,\\
\dot y=l(x-1)(x-\alpha)+2mxy +l\alpha y^2,
\end{gather*}
for which
$$
\mu_0=4\alpha(gm-hl)^2\ne0,\quad
C_2=-lx^3+(g-2m)x^2y+ (2h-l\alpha)xy^2+g\alpha y^3.
$$
In the same way as above using the factorization \eqref{factoriz:C2}
and solving the corresponding linear system with
respect to the parameters $g,h,l$ and $m$ we obtain
$$
g=\frac{V}{\alpha}, \quad h=\frac{V_u-\alpha U}{2},\quad
m=\frac{V-\alpha U_v}{2\alpha},\quad l=-U.
$$
Thus these parameters depend linearly on $U$, $U_v$, $V_u$ and $V$ and
applying the same arguments and actions as above we again obtain the
family of systems which depend exclusively on the coordinates of
their singular points (finite and infinite).
In short the Main Theorem is proved for systems with two real simple and
two complex singular points.
\subsection{Systems with four distinct complex singular points}
First we shall prove the following lemma concerning the complex singular
points of quadratic systems.
\begin{lemma}\label{lem:ImSP}
If a quadratic system \eqref{sys:gen} with real coefficients
possesses two complex (conjugated) singular points
$M_{1,2}(\alpha\pm { i }\beta,\gamma\pm { i }\delta)$ then via an
admissible affine transformation this system can be brought to the
form
\begin{equation}\label{CS:c1c1XX}
\begin{gathered}
\dot x= a +cx + gx^2 +2 h xy + ay^2,\\
\dot y= b +ex + lx^2 +2 m xy +b y^2,
\end{gathered}
\end{equation}
having the singular points $\widetilde M_{1,2}(0,\pm i)$.
\end{lemma}
\begin{proof}
Admit that a non-degenerate system \eqref{sys:gen} possesses the
singular points $M_{1,2}(\alpha\pm { i }\beta,\gamma\pm { i }\delta)$
($\beta^2+\delta^2\ne0$). Via the change $x\leftrightarrow y$ we can
assume $\beta\ne0$ and then we can consider $\beta=1$ via the rescaling $x\to
x/\beta$. Therefore the affine transformation
$$
\tilde x= -\delta x+y+\alpha\delta-\gamma,\quad \tilde y=x-\alpha,
$$
replace the singular points $M_{1,2}(\alpha\pm { i },\gamma\pm { i }\delta)$
by the singular points $\widetilde M_{1,2}(0,\pm { i })$.
Then since $P(0,{ i } )= Q(0,{ i } )=0$ yield $a_{00}+{ i }a_{01}-a_{02}=0$
and $b_{00}+{ i }b_{01}-b_{02}=0$. Thus $a_{01}=b_{01}=0$, $a_{02}=a_{00}$
and $b_{02}=b_{00}$ and setting some new parameters we obtain the canonical
system (\ref{CS:c1c1XX}).
\end{proof}
We shall construct now the canonical form of the family of quadratic systems
which possess four distinct complex singular points.
By Lemma \ref{lem:ImSP} doing an affine transformation we can locate
two complex singularities at the points $M_{1,2}(0,\pm { i })$.
So we shall consider systems (\ref{CS:c1c1XX}) which
besides the singular points $(0,\pm i)$ have the singular points
$M_{3,4}(x_{3,4},y_{3,4})$ where
$$
x_{3,4}=A\pm {i} B,\quad y_{3,4}=C\pm {i} D,\quad B^2+D^2\ne0,
$$
which are also complex. We claim that $x_{3,4}\ne0$, i.e. $A^2+B^2\ne0$.
Indeed, if $A=B=0$ then the point $(0,C+ {i} D)$
is a singular point of system (\ref{CS:c1c1XX}) and we have
\begin{gather*}
P(x_3,y_3)=a+a(C+ {i} D)^2= a(1+C^2-D^2+2 {i}CD)=0,\\
Q(x_3,y_3)=b+b(C+ {i} D)^2= b(1+C^2-D^2+2 {i}CD)=0.
\end{gather*}
Since $C,D\in \mathbb{R}$ and $a^2+b^2\ne0$ (otherwise systems
(\ref{CS:c1c1XX}) become degenerate) we obtain $C=0$ and $D=\pm
1$. Hence $(x_{3,4},y_{3,4})=(0,\pm i)$ and the complex singular points
have multiplicity $2$. This proves our claim.
We note that the transformation $x_1=\alpha x$, $y_1=\beta x+y$ keeps the
singular points $(0,\pm i)$ and transforms the singular points
$(x_{3,4},y_{3,4})$ to the points
\begin{equation}\label{SP}
\left(A\alpha \pm {i} B \alpha,\; A\beta+C\pm {i}(B\beta+D)\right)
\end{equation}
Since $A^2+B^2\ne0$ we shall consider two cases $B\ne0$, and $B=0, A\ne0$.
\subsubsection{The case $B\ne0$}
Then we may set $\alpha=1/B$,
$\beta=-D/B$ and the singular points (\ref{SP}) become $(p\pm {i},q)$
$(p,q\in \mathbb{R})$. In this case the relations $P(p\pm {i},q)= Q(p\pm {i},q)=0$
yield
\begin{gather*}
a(1+q^2)+ g(p^2-1)+ p( c + 2 h q)\pm {i}(c + 2g p + 2h q)=0,\\
b(1+q^2)+ l(p^2-1)+ p( e + 2 m q)\pm {i}(e + 2l p + 2m q)=0.
\end{gather*}
Herein we obtain the relations $a=g(p^2+1)/(q^2+1)$, $c=-2(gp+hq)$,
$b=l(p^2+1)/(q^2+1)$, $e=-2(lp+mq)$, and this leads to the following
family of systems
\begin{gather*}
\dot x= \frac{g(p^2+1)}{q^2+1}(1+y^2) -2(gp+hq)x + gx^2 +2 h xy,\\
\dot y= \frac{l(p^2+1)}{q^2+1}(1+y^2) -2(lp+mq)x + lx^2 +2 m xy,\\
\end{gather*}
with the singular points $M_{1,2}(0,\pm {i})$ and $M_{3,4}(p\pm {i},q)$.
For these systems the condition $gm-lh\ne0$ must be fulfilled (otherwise systems
become degenerate) and calculations yield
$$
C_2=-lx^3+(g-2m)x^2y +\Big[2h-l\,\frac{p^2+1}{q^2+1}\Big]xy^2
+g\,\frac{p^2+1}{q^2+1}y^3.
$$
Using the factorization \eqref{factoriz:C2} and solving the corresponding
linear system with respect to the parameters $g,h,l$ and $m$ we obtain
\begin{gather*}
g=\frac{q^2+1}{p^2+1}\,V,\quad m= \frac{(q^2+1)V-(p^2+1)U_v}{2(p^2+1)},\\
h=\frac{(q^2+1)V_u-(p^2+1)U}{2(q^2+1)},\quad l=-U.
\end{gather*}
Thus taking into consideration Lemma \ref{lem:C2-ISPs} we again get a family
of systems which depend exclusively on the coordinates of their singular
points (finite and infinite).
\subsubsection{The case $B =0$}
Then $A\ne0$ and from (\ref{SP}) by
setting $\alpha=1/A$ and $\beta=-C/A$ we obtain the singular points $(1,\pm
{i}p)$ with $p=D\ne0$. In this case the identities
$P(1,\pm {i}p)=Q(1,\pm {i}p)=0$ yield
$$
a(1-p^2)+ c+g \pm 2 {i}hp=0,\quad b(1-p^2)+ e+l \pm 2 {i}mp=0.
$$
Herein we obtain the relations
$$
h=m=0,\quad c= a(p^2-1)-g,\quad e= b(p^2-1)-l,
$$
which lead to the systems
\begin{gather*}
\dot x = a +[a(p^2-1)-g] x + gx^2 + ay^2,\\
\dot y = b +[b(p^2-1)-l] x + lx^2 + b y^2\\
\end{gather*}
with $al-bg\ne0$. These systems possess the singular points
$M_{1,2}(0,\pm {i})$, $M_{3,4}(1,\pm {i}p)$ and we calculate
$C_2=-lx^3+gx^2y -bxy^2 +ay^3$.
Considering the factorization \eqref{factoriz:C2} we obtain
$a=V$, $b=-V_u$, $g=U_v$, $l= -U$. Thus considering
Lemma \ref{lem:C2-ISPs} we obtain a family of systems which depends
exclusively on the coordinates of their singular points
(finite and infinite).
%\end{proof}
\section{Quadratic systems with three distinct finite singularities}
\subsection{Systems with one double and two simple real finite singular
points} Assume that a quadratic system \eqref{sys:gen} possesses
one double and two simple real singular points. By
\cite{Art_Llib_Vlp} in this case it can be brought via an
admissible (in the sense of Remark \ref{rem:admissible-trans})
affine transformation to the form
\begin{equation}\label{CS:r2r1r1}
\begin{gathered}
\dot x=cx+cq y - c\,x^2 +2 hxy - cq y^2,\\
\dot y=ex+eq y - e\,x^2 +2 mxy - eq y^2,
\end{gathered}
\end{equation}
with a double singular point $M_1(0,0)$ and two simple ones $M_2(1,0)$ and
$M_3(0,1)$. For these systems we calculate
$$
\mu_0=4q(cm-eh)^2\ne0,\quad C_2=ex^3-(c+2m)x^2y+(2h+eq)xy^2-cq y^3.
$$
Using the factorization \eqref{factoriz:C2} and solving the corresponding
linear system with respect to the parameters $c,e,h$ and $m$ we
obtain
\begin{equation}\label{val:param-r2r1r1}
c=-\frac{V}{q},\quad e=U,\quad h=\frac{V_u-qU}{2},\quad
m= \frac{V-qU_v}{2q}.
\end{equation}
Taking into consideration Lemma \ref{lem:C2-ISPs} we conclude that systems
\eqref{CS:r2r1r1} with parameters $c,e,h,m$ defined in
\eqref{val:param-r2r1r1}, form a family of systems which depends on the
coordinates of the infinite points (which can be complex and multiple) as well
as of the parameter $q$.
\subsection{Systems with one double and two simple complex singular points}
Assume that a quadratic system \eqref{sys:gen} possesses one
double and two complex singular points. By
\cite{Art_Llib_Vlp} in this case it can be brought via an
admissible affine transformation to the form
\begin{equation}\label{CS:r2c1c1a}
\begin{gathered}
\dot x= cm\,x +cn\,y +g\, x^2- cn\,xy + (g+cm)\,y^2,\\
\dot y= em\,x +en\,y +l\, x^2- en\,xy + (l+em)\,y^2.
\end{gathered}
\end{equation}
with a double singular point $M_1(0,0)$ and two complex $M_{2,3}(1,\pm i)$.
For these systems we calculate
\begin{gather*}
\mu_0=(cl-eg)^2(m^2+n^2)\ne0,\\
C_2=-lx^3+(g+en)x^2y-(l+em+cn)xy^2+(g+cm) y^3.
\end{gather*}
Considering the factorization \eqref{factoriz:C2} we obtain
\begin{equation}\label{val:param-r2c1c1}
\begin{gathered}
c=\frac{nU+mV-mU_v-nV_u}{m^2+n^2},\quad
e=\frac{mU-nV+nU_v-mV_u}{m^2+n^2},\\
g=\frac{n^2V-mnU+m^2U_v+mnV_u}{m^2+n^2},\quad
l= -U.
\end{gathered}
\end{equation}
Taking into account these relations we observe that all the
coefficients of systems \eqref{CS:r2c1c1a} are homogeneous
functions on the parameters $m$ and $n$. So, since $(m^2+n^2)\ne0$
considering Lemma \ref{lem:C2-ISPs} we conclude that the family
of systems \eqref{CS:r2c1c1a} with parameters $c,e,g$ and $l$
defined in \eqref{val:param-r2c1c1}, is a family which depends on
the coordinates of infinite points as well as of one parameter
$\gamma=m/n$ or $\gamma=n/m$.
\subsection{Systems with three simple real finite singular
points}
In this case only one finite singular point is gone to
infinity (i.e. $\mu_0=0$) and hence the polynomials $p_2(x,y)$ and
$q_2(x,y)$ have a common linear factor. By Remark
\ref{rem:lin-fct-p2,q2} due to an admissible transformation we
can assume $a_{02}=b_{02}=0$ for systems \eqref{sys:gen}. Then
via a translation one of the finite singularities can be placed at
the origin and we obtain the systems
\begin{equation}\label{CS:r1r1r1a} \dot
x=cx+d y + gx^2 +2 hxy\equiv P(x,y),\quad
\dot y=ex+f y +lx^2 +2 mxy\equiv Q(x,y),
\end{equation}
with $M_1(0,0)$.
We claim that the other real singular points
$M_{2}(x_{2},y_{2})$ (and $M_{3}(x_{3},y_{3})$) of these systems
has the coordinate $x\ne0$. Indeed, if we suppose $x_2=0$ then we
obtain $\ P(0,y_2)=d\,y_2=0,\ $ $ Q(0,y_{2})=f\,y_{2}=0, $ and
since $y_{2}\ne0$ ($M_1$ and $M_{2}$ are distinct) we have
$d=f=0$. However the last relations yield degenerate systems.
Thus $x_2\ne0$ and via the linear transformation $
\bar x ={x}/{x_2}$ and either $\bar y =y $
if $ y_2=0$, {or} $ \bar y= x-{x_2}y/{y_2}$
{if} $y_2\ne0$
(which keeps the form \eqref{CS:r1r1r1a}), we locate the singular point
$M_2(x_2,y_2)$ at the point $M_2(1,0)$. In this way we obtain the
systems
\begin{equation}\label{CS:r1r1r1}
\dot x=cx+d y - cx^2 +2 hxy\equiv P(x,y),\quad
\dot y=ex+f y - ex^2 +2 mxy\equiv Q(x,y)
\end{equation}
with the three singular points $M_1(0,0)$, $M_2(1,0)$ and
$M_3(\alpha,\beta)$. Now will find the dependence among the
coefficients of system \eqref{CS:r1r1r1} and the parameters
$\alpha$ and $\beta$. Since $\beta\ne0$ (we cannot have three distinct singular
points placed on the line $y=0$) and
$P(\alpha,\beta)=Q(\alpha,\beta)=0$, we obtain
$$
d=\frac{c\alpha(\alpha-1)-2h\alpha\beta}{\beta},\quad
f=\frac{e\alpha(\alpha-1)-2m\alpha\beta}{\beta}.
$$
Therefore after the time rescaling ($t\to \beta t_1$) and the
re-parametrization ($h\beta\to h$, $m\beta\to m$) we get the
following family of systems
\begin{equation}\label{CS:r1r1r1-a}
\begin{gathered}
\dot x=c\,\beta x(1-x)+c\,\alpha(\alpha-1)y +2h(x-\alpha)y,\\
\dot y=e\,\beta x(1-x)+e\,\alpha(\alpha-1)y +2m(x-\alpha)y.
\end{gathered}
\end{equation}
Evidently each system of this family possesses the singular
points $M_1(0,0)$, $M_2(1,0)$ and $M_3(\alpha,\beta)$ and for
this family according to Lemma \ref{lem:mu_i-Infty} we have
$\mu_0=0$ and the following condition is satisfied $
\mu_1=4(cm-eh)^2\alpha\beta(1-\alpha)x\ne0$,
otherwise the systems become degenerate. For these systems we
calculate $C_2=e\beta x^3-(c\beta+2m)x^2y+2hxy^2$.
Using the factorization \eqref{factoriz:C2} (in this case $V=0$)
and solving the corresponding linear system with respect to the
parameters $c,e,h$ and $m$ we can determine only three parameters
$e,h$ and either $c$ or $m$. So we get $e=U/\beta$, $h=V_u/2$,
$c= -(2m+U_v)/\beta$.
Therefore considering Lemma \ref{lem:C2-ISPs} we conclude that
the family of systems \eqref{CS:r1r1r1-a} becomes a family which
depends on the coordinates of singular points (finite and
infinite) as well as on the parameter $m$.
\subsection{Systems with one real simple and two complex finite singular
points}
In this case taking into account Remark \ref{rem:lin-fct-p2,q2} according
to \cite{Art_Llib_Vlp} via an admissible affine transformation the
quadratic systems can be written into the form
\begin{equation}\label{CS:r1c1c1}
\begin{gathered}
\dot x=-2(h+g q)x+g(q^2+1)y + gx^2 +2 hxy,\\
\dot y=-2(m+l q)x+l(q^2+1)y + lx^2 +2 mxy,
\end{gathered}
\end{equation}
with three singular points $M_1(0,0)$ and
$M_{2,3}(q\pm i,1)$. For these systems considering
Lemma \ref{lem:mu_i-Infty} we have $\mu_0=0$ and we calculate
$$
\mu_1=4(gm-hl)^2(q^2+1)x\ne0,\quad C_2=-l x^3+(g-2m)x^2y+2hxy^2.
$$
Then using the factorization \eqref{factoriz:C2} we can determine
only the three parameters $l,h$ and either $g$ or $m$. So we have
$g=2m+U_v,\quad h=V_u/2,\quad l=-U$.
By Lemma \ref{lem:C2-ISPs} the family of systems \eqref{CS:r1c1c1}
becomes a family which depends on the coordinates of singular
points (finite and infinite) as well as on the parameter $m$.
\section{Quadratic systems with two distinct finite singularities}
\subsection{Systems with two double real finite singular points}
By \cite{Art_Llib_Vlp} and doing an affine transformation
(using only the coordinates of the two double real singularities)
a such quadratic system can be written into the form
\begin{equation}\label{CS:r2r2}
\begin{gathered}
\dot x=cx+cpy -cx^2 +2cqxy+ky^2,\\
\dot y=ex+epy -ex^2 +2 eqxy+ny^2,
\end{gathered}
\end{equation}
with two double singular points $M_1(0,0)$ and $M_{2}(1,0)$.
For these systems we calculate
$$
\mu_0=(cn-ek)^2\ne0,\quad C_2=e x^3-(c+2eq)x^2y+(2cq-n)xy^2+ky^3.
$$
Then using the factorization \eqref{factoriz:C2} and solving the
corresponding system of equations with respect to the parameters $c,e,n$ and
$k$ we obtain $c=-(2qU+U_v)$, $e=U$, $k=V$, $n=-(4q^2U+2qU_v+V_u)$.
Considering Lemma \ref{lem:C2-ISPs} we get that the family of systems
\eqref{CS:r2r2} becomes a family which depends on the coordinates of infinite
singular points as well as on two independent parameters $p$ and $q$.
\subsection{Systems with two double complex finite singular points}
First we shall construct the respective canonical form for this class of
systems. Assume that a quadratic system possesses 2 complex singular
points. Then according to \cite{Art_Llib_Vlp} (see the proof of Lemma 4.3)
these points can be replaced by the points $ M_{1,2}(0,\pm { i })$. Thus we
consider the canonical system
\begin{equation}\label{CS:c1c1XXa}
\begin{gathered}
\dot x= a +cx + gx^2 +2 h xy + ay^2,\\
\dot y= b +ex + lx^2 +2 m xy +b y^2,
\end{gathered}
\end{equation}
which besides the singular points $(0,\pm i)$ has the singular points
$M_{3,4}(x_{3,4},y_{3,4})$ where
\begin{gather*}
x_{3,4}=\left(2d_{25}d_{56}-d_{26}d_{46}\pm 2d_{56}\sqrt{\tilde
D}\,\right)\big/ \mu_0,\\
y_{3,4}=\left(d_{26}d_{45}-d_{24}d_{56}\mp
d_{46}\sqrt{\tilde D}\,\right)\big/
\mu_0,
\end{gather*}
and $\tilde D=d_{25}^2-d_{46}^2-d_{24}d_{26}+4d_{45}d_{56}$,
$\mu_0=d_{46}^2-4d_{45}d_{56}\ne0$ (by Lemma \ref{lem:mu_i-Infty})
and
\begin{gather*}
d_{24}=cl-eg,\quad d_{25}=cm-eh,\quad d_{26}=bc-ae,\\
d_{45}=gm-hl,\quad d_{46}=bg-al,\quad d_{56}=bh-am.
\end{gather*}
In order to have two double complex singular points it is necessary
that $\tilde D<0$. Therefore we can have $x_{3}=x_{4}=0$ if
and only if $d_{56}=d_{26}d_{46}=0$. Since the condition
$\mu_0\ne0$ imply $d_{46}\ne0$ we obtain $bc-ae=bh-am=0$.
Due to the condition $a^2+b^2\ne0$ (otherwise systems (\ref{CS:c1c1XXa})
become degenerate) we may set $c=a p,\ e=b p$ and $ h=a q,\ m=bq$
(where $p$ and $q$
are some new parameters). Then we obtain the needed family of quadratic
systems
\begin{equation}\label{CS:c2c2a}
\begin{gathered}
\dot x= a +a px + gx^2 +2 aq xy + ay^2,\\
\dot y= b +b px + lx^2 +2 bq xy + b y^2,
\end{gathered}
\end{equation}
possessing two double complex singular points $ M_{1,2}(0,\pm { i })$.
For these systems we calculate
$$
\mu_0=(al-bg)^2\ne0,\quad C_2=-l x^3+(g-2bq)x^2y+(2aq-b)xy^2+ay^3.
$$
Then using the factorization \eqref{factoriz:C2} and solving the
corresponding system of linear equations with respect to the
parameters $c,e,n$ and $k$ we obtain
$$
a=V,\quad b=2qV-V_u,\quad g=4q^2V+U_v-2qV_u,\quad l=-U.
$$
By Lemma \ref{lem:C2-ISPs} we obtain that the family of systems
\eqref{CS:c2c2a} becomes a family which depends on the coordinates
of the infinite singular points (finite points being fixed) as well
as on two independent parameters $p$ and $q$.
\subsection{Systems with one triple and one simple real finite
singular points}
By \cite{Art_Llib_Vlp} in this case via an affine
transformation (depending only on the coordinates of the two
finite singularities) a such quadratic system can be written into the
form
\begin{equation}\label{CS:r3r1}
\begin{gathered}
\dot x=cx+cpy -cx^2 +2hxy+(2hp+cq)y^2,\\
\dot y=ex+epy -ex^2 +2 mxy+(2mp+eq)y^2,
\end{gathered}
\end{equation}
with one triple $M_1(0,0)$ and one simple $M_{2}(1,0)$ real singular points.
For these systems we calculate
\begin{gather*}
\mu_0=4(p^2-q)(cm-eh)^2\ne0,\\
C_2=e x^3-(c+2m)x^2y+(2h-2mp-eq)xy^2+(2hp+cq)y^3.
\end{gather*}
Using the factorization \eqref{factoriz:C2} and solving the corresponding
linear system with respect to the parameters $c,e,m$ and $h$ (as
$q-p^2\ne0$) we obtain
\begin{gather*}
c=\frac{-p\,qU+p^2U_v-pV_u+V}{q-p^2},\quad
h = \frac{q^2U-p\,qU_v+qV_u-pV}{2(q-p^2)},\\
m = \frac{p\,qU-qU_v+pV_u-V}{2(q-p^2)}, \quad e=U.
\end{gather*}
Taking into account Lemma \ref{lem:C2-ISPs} we obtain that
the family of systems \eqref{CS:r3r1} becomes a family which depends
on the coordinates of the infinite singular points (finite singular
points being fixed) as well as on two independent parameters $p$ and $q$.
\subsection{Systems with one double and one simple real finite
singular points}
Since in this case only one finite singularity has gone to infinity
we conclude that $p_2(x,y)$ and $q_2(x,y)$ have a linear common factor.
Then taking into consideration Remark \ref{rem:lin-fct-p2,q2},
according to \cite{Art_Llib_Vlp} via an admissible affine
transformation a such quadratic system can be written into the form
\begin{equation}\label{CS:r2r1}
\begin{gathered}
\dot x=cx+cpy -cx^2 +2hxy,\\
\dot y=ex+epy -ex^2 +2 mxy,
\end{gathered}
\end{equation}
with one double $M_1(0,0)$ and one simple $M_{2}(1,0)$ real
singular points. By Lemma \ref{lem:mu_i-Infty} we have
$\mu_0=0$ and the condition $ \mu_1=-4p(cm-eh)^2x\ne0 $ holds. We
calculate $C_2=e x^3-(c+2m)x^2y+2hxy^2$.
Then using the factorization \eqref{factoriz:C2} and solving the
corresponding linear system with respect to the parameters $c,e$
and $h$ we obtain $c= - (2m+U_v)$, $e={U}$, $h= V_u/2$.
By Lemma \ref{lem:C2-ISPs} we obtain that the family of systems
\eqref{CS:r2r1} becomes a family which depends on the coordinates
of the infinite singular points as well as on two independent
parameters $m $ and $p$.
\subsection{Systems with two simple real finite singular points}
\label{section:2real}
In this case $\mu_0=\mu_1=0$ and two singular points of quadratic
systems (\ref{sys:gen}) have gone to infinity. Therefore the form
of the respective canonical systems depends of the degree
of $\gcd(p_2,q_2)$ and we shall investigate two cases $K\ne0$ and $K=0$.
\subsubsection{The case $K\ne0$.} Then $\deg(\gcd(p_2,q_2))=1$,
i.e. $p_2(x,y)$ and $q_2(x,y)$ have a linear common factor.
So taking into consideration Remark \ref{rem:lin-fct-p2,q2},
according to \cite{Art_Llib_Vlp} via an admissible affine
transformation a quadratic system in this case can be written into
the form
\begin{equation}\label{CS:r1r1a}
\begin{gathered}
\dot x=cx+d y - cx^2 +2 dq xy,\\
\dot y= ex+f y - ex^2 +2 fq xy,
\end{gathered}
\end{equation}
which possess simple singular points $M_1(0,0)$ and $M_2(1,0)$.
For these systems we calculate $\mu_0=\mu_1=0$,
$\mu_2=(cf-de)^2(2q+1)x^2\ne0$, $K= q(de-cf)x^2\ne0$ and
$ C_2=e x^3-(c+2fq)x^2y+2d qxy^2$.
Then using the factorization \eqref{factoriz:C2} and solving
the corresponding system of linear equations
with respect to the parameters $c,e$ and $d$ (since $q\ne0$)
we get the following relations $c= -2fq-U_v$, $e=U$, $d= V_u/(2q)$.
Thus the family of systems \eqref{CS:r1r1a} becomes a family which
depends on the coordinates of the infinite singular points as well
as on two independent parameters $f$ and $q$.
\subsubsection{The case $K=0$.} \label{subsect:2realK=0}
In this case the polynomials $p_2(x,y)$ and $q_2(x,y)$ are
proportional, i.e. the identity $\alpha p_2(x,y)-\beta q_2(x,y)=0$
with $\alpha^2+\beta^2\ne0$ holds in $\mathbb{R}[x,y]$. Moreover we can
assume $\alpha\ne0$ due to the change $x\leftrightarrow y$.
Since $C_2(a,x,y)=y p_2(x,y)-x q_2(x,y)$ we conclude that one of
the factors of $C_2$ in the factorization \eqref{factoriz:C2}
coincides with $\alpha x-\beta y$. If $M_1(x_0,y_0)$ is a singular
point of the quadratic systems, then via the admissible affine
transformation $x_1= \alpha x-\beta y-x_0$ and $y_1=y-y_0$ we
obtain the systems (keeping the old notation)
\begin{gather*}
\dot x= cx+ d y\equiv P(x,y),\\
\dot y= ex+ f y +lx^2 +2 m x y +ny^2\equiv Q(x,y).
\end{gather*}
Besides the singular point $M_1(0,0)$ these systems
possess a real simple singular point $M_2(x_2,y_2)$. Since
$x_2^2+y_2^2\ne0$, using the transformation $x_1=x$ and either
$y_1=y/y_2$ (if $y_2\ne0$), or $y_1=x/x_2+y$ (if $y_2=0$) the point
$M_2$ could be placed at the coordinates $(q,1)$, where $q$ is
some parameter. So, since $P(q,1)=Q(q,1)=0$
we get the family of systems
\begin{equation}\label{CS:r1r1c}
\begin{gathered}
\dot x = c(x-q y),\\
\dot y = e(x-qy)-(n+2mq+lq^2)y +lx^2 +2 m x y +ny^2,
\end{gathered}
\end{equation}
for which $\mu_0=\mu_1=0$, $\mu_2=c^2(n+2mq+lq^2)q_2(x,y)\ne0$
and $C_2=-l x^3-2m x^2y-nxy^2$.
Then using the factorization \eqref{factoriz:C2} and solving the
corresponding linear system with respect to the
parameters $l,m$ and $n$ we get the following relations
$l =-U$, $m=-U_v/2$, $n=-V_u$. Thus the family of systems
\eqref{CS:r1r1c} becomes a family which depends on the coordinates
of the singular points (finite and infinite) as well as on two
independent parameters $e$ and $c$.
\subsection{Systems with two complex finite singular points
} We shall consider the canonical system (\ref{CS:c1c1XX}) which
has the singular points $(0,\pm i)$ and some two other singular
points $(x_{3,4},y_{3,4})$. To construct the needed canonical form
we must find out the conditions on the parameters which locate
both singular points $(x_{3,4},y_{3,4})$ at infinity. For this
according to Lemma \ref{lem:mu_i-Infty} the conditions
$\mu_0=\mu_1=0$ have to be fulfilled.
For system (\ref{CS:c1c1XX}) we calculate
\begin{equation}\label{val:2}
\mu_0= d_{46}^2-4 d_{45}d_{56}=0,\quad
K=d_{45}x^2+d_{46}xy+d_{56}y^2,
\end{equation}
where $d_{45}=gm-hl,\ d_{46}=gb-al,\ d_{56}=bh-am$ and we shall
consider two cases $K\ne0$ and $K=0$.
\subsubsection{The case $K\ne0$.}
Since $\mu_0$ is the discriminant of the binary form
$K(a,x,y)\ne0$ then the condition $\mu_0=0$ implies
$K(a,x,y)=\pm(\alpha x+\beta y)^2\ne0$.
On the other hand, computations yield that
$\mathop{\rm Resultant}[K(a,x,y), C_2(x,y),\gamma]=\mu_0 W(a)$, $\gamma=x/y$ or $\gamma=y/x$,
where $W(a)$ is a polynomial of degree 4 in the coefficients of
systems \eqref{sys:gen}. Hence the condition $\mu_0=0$ also
implies that $K$ and $C_2$ have a common non--constant factor,
which in this case is $\alpha x+\beta y$. Hence this factor
indicates a real infinite singular point and we claim that this
point cannot be in the direction $x=0$, i.e. $\beta\ne0$.
Indeed suppose $\beta=0$. Then $K=\tilde\alpha x^2$ and from
\eqref{val:2} we obtain $d_{56}=d_{46}=0$, $d_{45}\ne0$. Therefore
we obtain the relations $d_{45}=gm-hl\ne0$, $d_{46}=gb-al=0$,
$d_{56}=bh-am=0$, which imply $a=b=0$. This leads to degenerate systems
(\ref{CS:c1c1XX}). So our claim is proved and $\beta\ne0$.
Now via the admissible transformation $ x_1=x$, $
y_1=\alpha x/\beta+y $ (which keeps the singular points $(0,\pm
i)$ and depends only on the coordinates of the infinite points) we
obtain $K=d_{56}y^2\ne0$. So we have reached the relations
$d_{45}=d_{46}=0$ and hence we have $d_{45}=gm-hl=0$, $d_{46}=gb-al=0$,
$d_{56}=bh-am\ne0$. Herein we obtain $g=l=0$. Then for system
(\ref{CS:c1c1XX}) we calculate $\mu_1=-4 (b h - a m) (e h - c m) y$
and the condition $\mu_1=0$ implies $eh-cm=0$. Since the condition
$d_{56}=bh-am\ne0$ yields $m^2+h^2\ne0$ we may set $c=hq$ and $e=mq$.
Thus we get the following systems
\begin{equation}\label{CS:c1c1a}
\begin{gathered}
\dot x = a +hq x + 2 h xy + ay^2,\\
\dot y = b +mq x + 2 m xy + b y^2,
\end{gathered}
\end{equation}
which have the singular points $M_{1,2}(0,\pm {i})$ and the other
two singular points have gone to infinity. For these systems we calculate
$\mu_0=\mu_1=0$, $\mu_2=(am-bh)^2(q^2+4)y^2\ne0$ and $C_2=-2mx^2y+ (2h-b)xy^2+ay^3$. Then using the factorization \eqref{factoriz:C2} and solving the corresponding linear system with respect to the parameters $a,b$ and $m$ we get $a =V$, $b=2h-V_u$, $m=-U_v/2$. Thus the family of systems
\eqref{CS:c1c1a} becomes a family which depends on the coordinates
of the singular points (finite and infinite) as well as on two
independent parameters $h$ and $q$.
\subsubsection{The case $K=0$}
In this case the polynomials $p_2(x,y)$ and $q_2(x,y)$ are
proportional and as above (see subsection \ref{subsect:2realK=0})
via an admissible linear transformation we can force
$a_{20}=a_{11}=a_{02}=0$. Therefore we obtain the systems
\begin{gather*}
\dot x = a+cx+ d y\equiv P(x,y),\\
\dot y = b+ ex+ f y +lx^2 +2 m x y +ny^2\equiv Q(x,y),
\end{gather*}
which need to possess two complex singular points
$M_{1,2}(A\pm i B ,C\pm i D)$ with $B^2+D^2\ne0$, and we shall
consider two subcases $B\ne0$ and $B=0$.
{\it Subcase $B\neq 0$}. Using the transformation $x_1=x/B$ and
$y_1=-Dx+By $ we place these points at $\tilde
M_{1,2}(p\pm i, q)$ where $p$ and $q$ are some parameters. So, since
$P(p+ i,q)=Q(p+ i,q)=0$ we get the family of systems
\begin{equation}\label{CS:c1c1d-1}
\begin{gathered}
\dot x= d(y-q),\\
\dot y= l(p^2+1)-q(f+nq) -2(lp+mq)x+f y +lx^2 +2 m x y +ny^2,
\end{gathered}
\end{equation}
for which $\mu_0=\mu_1=0$, $\mu_2=d^2l(lx^2 +2 m x y +ny^2)\ne0$.
For these systems we calculate $C_2=-l x^3-2m x^2y-nxy^2$.
Then using the factorization \eqref{factoriz:C2} and solving
the corresponding linear system with respect to the parameters
$l,m$ and $n$ we get $l =-U$, $m=-U_v/2$ and $n=-V_u$.
Thus the family of systems \eqref{CS:c1c1d-1} becomes a family
which depends on the coordinates of the singular points
(finite and infinite) as well as on two independent parameters
$d$ and $f$.
{\it Subcase $B=0$}. Then $D\ne0$ and via the rescaling
$y\to y/D$ we get the singular points $\tilde M_{1,2}(p, q\pm i)$ where
$p=A$ and $q=C/D$. So, from $P(p,q+i)=Q(p,q+i)=0$ we get the family of systems
\begin{equation}\label{CS:c1c1d-1a}
\begin{gathered}
\dot x= c(x-p),\\
\dot y= n(q^2+1)-p(e+lp)+ex-2(mp+nq) y +lx^2 +2 m x y +ny^2,
\end{gathered}
\end{equation}
for which $\mu_0=\mu_1=0$, $\mu_2=c^2n(lx^2 +2 m x y +ny^2)\ne0$.
For these systems we obtain $C_2=-l x^3-2m x^2y-nxy^2$,
and taking into consideration the factorization
\eqref{factoriz:C2} and solving the corresponding linear system with
respect to the parameters $l,m$ and $n$ we get $l =-U$, $m=-U_v/2$
and $n=-V_u$. Thus the family of systems \eqref{CS:c1c1d-1a}
becomes a family which depends on the
coordinates of the singular points (finite and infinite) as well as
on two independent parameters $c$ and $e$.
\section{Systems with at most one finite singular point}
In this section we shall use another point of view. Since this
family of systems has at most one finite singularity, in order
to use admissible (in the sense of Remark \ref{rem:admissible-trans})
affine transformations we shall use the possible configurations of
infinite singular points.
\subsection{The case of three real infinite singular points} In this case
the polynomial $C_2=yP(x,y)-xQ(x,y)$ has three real linear
factors. Therefore via a linear transformation this binary cubic
form can be written in the form $C_2=xy(x-y)$. We note that the
applied transformation is admissible (see Remark \ref{rem:admissible-trans}) because it depends only on the coordinates of infinite singularities of systems \eqref{sys:gen}. Then using the factorization \eqref{factoriz:C2} and a time rescaling we get the systems
\begin{equation}\label{CS:Inf-S1-0}
\dot x= a+cx+dy+gx^2+(h-1)xy,\quad
\dot y=b+ex+fy+(g-1)xy+hy^2,
\end{equation}
with $\mu_0=gh(g+h-1)$.
\subsubsection{The case $\mu_0\ne0$} By Lemma \ref{lem:mu_i-Infty}
the unique finite singularity must be of multiplicity four.
Translating this point to the origin of coordinates we get the
family of systems
\begin{equation}\label{CS:Inf-SS1}
\dot x= cx+dy+gx^2+(h-1)xy,\quad
\dot y= ex+fy+(g-1)xy+hy^2.
\end{equation}
Clearly if the singular point $(0,0)$ has at least multiplicity 2
the condition $cf-de=0$ must hold.
\noindent \textit{Subcase $d\ne0$}. Then $e=cf/d$ and for systems
\eqref{CS:Inf-SS1} calculations yield
$$
\mu_4=\mu_3=0,\quad \mu_2=\frac{\mathcal{F}_1}{d}
\big[fgx^2+(d-f-dg+fh)xy-dhy^2\big],
$$
where $ \mathcal{F}_1=f(dg+c-ch)-c(d-dg+ch)$.
By Lemma \ref{lem:mu_i-(0,0)} in order to have a point of
multiplicity 4 we must force the conditions $\mu_2=\mu_1=0$
to be fulfilled. Since $\mu_0\ne0$ (i.e. $h\ne0$) the condition
$\mu_2=0$ is equivalent to $ \mathcal{F}_1=0$.
We claim that for $dg+c-ch=0$ we cannot have a point of multiplicity 4.
Indeed, if $g=c(h-1)/d$ and we get the contradiction
$\mu_0=c(c+d)h(h-1)^2/d^2\ne0$
and $\mathcal{F}_1=-c(c+d)=0$. So $dg+c-ch\ne0$ and the condition
$\mathcal{F}_1=0$ gives $f=c(d-dg+ch)/(dg+c-ch)$. Then we calculate
$$
\mu_1=\frac{\mathcal{F}_2}{d(dg+c-ch)}\big[g(dg-d-ch)x+h(dg+c-ch)y\big],
$$
where\quad $ \mathcal{F}_2=(dg-ch)^2-c^2h-d^2g$. Since $h(dg+c-ch)\ne0$ the
condition $\mu_1=0$ is equivalent to $ \mathcal{F}_2=0$. We observe that
$\mathop{\rm Discrim}[\mathcal{F}_2,c]=4d^2gh(g+h-1)=4d^2\mu_0$. So for the existence of
real parameters $c,d, h$ and $g$ in order to have a point of
multiplicity 4 for a non--homogeneous quadratic system \eqref{sys:gen}
it is necessary $\mu_0>0$. Then we have
$c_{1,2}= (dgh\pm d\sqrt{gh(g+h-1)})/(h(h-1))= d\tilde c_{1,2}$,
and this leads to the families of systems
\begin{gather*}
\dot x= dc_0x+dy+gx^2+(h-1)xy,\\
\dot y= \frac{ dc_0^2(1-g+c_0h)}{c_0+g-c_0h}x
+\frac{ dc_0 (1-g+c_0h)}{c_0+g-c_0h}y+(g-1)xy+hy^2,
\end{gather*}
where $c_0$ is one of the values $\tilde c_{1,2}$.
Therefore we get two families of systems, each of them depending
on 3 parameters. However fixing the coordinates of three
distinct real infinite singularities and fixing the multiplicity 4
for finite singularity we automatically get only one of these
families, which depends on three parameters.
\noindent \textit{Subcase $d=0$}.
Then we have $cf=0$ and we claim that in order to have the singular
point $M_0(0,0)$ of multiplicity 4 it is necessary that $c=0$.
Indeed if $c\ne0$ then $f=0$ and for systems \eqref{CS:Inf-SS1}
we get $\mu_4=\mu_3=0$ and $\mu_2=ch\big[-egx^2+(e-c+cg-eh)xy+chy^2\big]$.
Since $\mu_0\ne0$ (i.e. $h\ne0$) we obtain $ch\ne0$ and this yields
$\mu_2\ne0$.
Now $c=0$ and for systems \eqref{CS:Inf-SS1} we calculate $\mu_4=\mu_3=0$
and $\mu_2=f(e+fg-eh) \big[gx +(h-1)y\big]x$.
In this case the condition $\mu_2=0$ implies $f(e+fg-eh)=~0$.
Then calculations yield either $\mu_1=eh(1-h)\big[gx
+(h-1)y\big]$ if $f=0$, or $\mu_1=e(h-1)(g+h-1)x$ if $f=e(h-1)/g$.
Since $\mu_0\ne0$ in both cases we get $f=0=e(h-1)$. Therefore in
the case $f=e=0$ as well as in the case $f=h-1=0$ we obtain a
family of quadratic systems depending on two parameters.
\subsubsection{Case $\mu_0=0$, $\mu_1\ne0$}
The condition $\mu_0=0$ yields
$gh(g+h-1)=0$ and without loss of generality we may assume that
for systems \eqref{CS:Inf-S1-0} the condition $g=0$ holds. Indeed,
if $h=0$ (respectively $g+h-1=0$) we can apply the admissible
linear transformation which sends the straight line $y=0$
to $ x=0$ (respectively $y=0 $ to $y=x$). So assuming $g=0$
and translating the singular point to the origin of coordinates we
get the family of systems
\begin{equation}\label{CS:Inf-SS1Triple}
\dot x= cx+dy +(h-1)xy,\quad
\dot y= ex+fy-xy+hy^2,
\end{equation}
for which $\mu_1=h(h-1)(e-eh-c)y\ne0$ because $(0,0)$ must be of
multiplicity 3. Clearly in order to have at least of multiplicity 2 the
condition $cf-de=0$ must hold.
\noindent \textit{Subcase $d\ne0$}. Then $e=cf/d$ and for systems
\eqref{CS:Inf-SS1} calculations yield $\mu_4=\mu_3=0$ and $\mu_2=c\big[f(1-h)-ch-d\big]\big[(fh-f+d)x - dhy\big]y/d$.
By Lemma \ref{lem:mu_i-(0,0)} to have a point of
multiplicity 3 we must force the condition $\mu_2=0$ to be
fulfilled. Since $\mu_1=ch(1-h)(fh-f+d)y/d\ne0$ (i.e. $ch\ne0$)
the condition $\mu_2=0$ is equivalent to $ f(1-h)-ch-d=0$ and this
gives $c=(f-fh-d)/h$. Thus we obtain the family depending on
three parameters $d$, $f$ and $h$.
\noindent \textit{Subcase $d=0$}. Then we have $cf=0$ and we claim that to in order to have the singular point $M_0(0,0)$ of multiplicity 3 it is necessary
$c=0$. Indeed, if $c\ne0$ then $f=0$ and for systems
\eqref{CS:Inf-SS1Triple} we get $\mu_4=\mu_3=0$ and $\mu_2=ch\big[(e-eh-c)x+ chy\big]y$. Since $\mu_1\ne0$ (i.e. $h\ne0$) we obtain $ch\ne0$ and this
yields $\mu_2\ne0$. Thus $c=0$ and in this case we calculate
$\mu_1=-eh(h-1)^2y$ and $\mu_2=-ef(h-1)^2xy$, and by $\mu_1\ne0$ the condition $\mu_2=0$ gives $f=0$. So we get the family of systems depending on two parameters $e$ and $h$.
\subsubsection{Case $\mu_0=\mu_1=0$}
In this case according to Lemma \ref{lem:mu_i-Infty} at least two
finite singularities have gone to infinity. As it was shown above
from $\mu_0=gh(g+h-1)=0$ without loss of generality we may
assume that for systems \eqref{CS:Inf-S1-0} the condition $g=0$
holds. Then for these systems calculations yield
\begin{equation}\label{val:kappa}
\mu_1=(c-e+eh)h(1-h)y,\quad \kappa=16h(1-h).
\end{equation}
By Lemma \ref{lem:inf-config} for $\mu_0=\mu_1=0$ the
configurations of infinite singularities are governed by the invariant
polynomial $\kappa(a)$.
{\it Subcase $\kappa\ne0$.} Then $h(h-1)\ne0$
and he condition $\mu_1=0$ yields $c=e(1-h)$. Thus we get the
4--parameter family of systems
\begin{equation}\label{CS:Inf-S1}
\dot x=a +e(1-h)x+dy + (h-1) xy,\quad \dot y=b +ex+fy-xy +hy^2,
\end{equation}
for which
\begin{equation}\label{val:mu2}
\mu_2=h(h-1)\big[a+de+(h-1)(b+ef+e^2h)\big]y^2.
\end{equation}
We separate the proof of this subcase in three pieces.
\noindent First $\mu_2\ne0$. In this case systems possess exactly
one real singular point of multiplicity $2$. Then translating this
point to the origin of coordinate we obtain the systems
\begin{equation}\label{CS:Inf-S1-1a}
\dot x= e(1-h)x+dy + (h-1) xy,\quad \dot y= ex+fy-xy +hy^2,
\end{equation}
for which $\mu_0=\mu_1=0$, $\mu_2=e h(h-1)\big[d +(h-1)(f+eh)\big]y^2$, $\mu_3=e(d-f+fh)\big[(d +(h-1)(f+eh))x-dhy\big]y^2$ and $\mu_4=0$.
By Lemma \ref{lem:mu_i-(0,0)} in order to have a double
point at the origin of coordinates we must force $\mu_3=0$. Since
$\mu_2\ne0$ we obtain $d=f(1-h)$ and we get the family of systems
$$
\dot x= (1-h)(ex+fy - xy),\quad \dot y= ex+fy-xy +hy^2,
$$
which depends on three parameters $e$, $f$ and $h$.
\noindent Second $\mu_2=0$ and $\mu_3\ne0$. In this case by Lemma
\ref{lem:mu_i-(0,0)} the singular point $M_0(0,0)$ of systems
\eqref{CS:Inf-S1-1a} is a simple real one. The conditions $\mu_2=0$
and $\mu_3\ne0$ imply $d =(1-h)(f+eh)$. Thus the family of systems
\eqref{CS:Inf-S1-1a} becomes again a family depending of three
parameters $e$, $f$ and $h$.
\noindent Third $\mu_2=\mu_3=0$ and $\mu_4\ne0$. Since there are no finite
singularities and $\mu_0=\mu_1=0$, we shall consider systems
\eqref{CS:Inf-S1} for which according to \eqref{val:mu2} the
conditions $\mu_2=0$ and $\kappa\ne0$ (i.e. $h(h-1)\ne0$) yield
$a=-de+(1-h)(b+ef+e^2h)$. Then we calculate
\begin{gather*}
\mu_3=h(1-h)(b+ef+e^2h)\big[d +(h-1)(f+2eh)\big]y^3,\\
\mu_4=(b+ef+e^2h)y^3\,W(b,e,f,h,x,y)
\end{gather*}
where $W(b,e,f,h,x,y)$ is a linear homogeneous polynomial in $x$
and $y$. Taking into consideration the condition $\kappa\mu_4\ne0$
the relation $\mu_3=0$ yields $d =(1-h)(f+2eh)$. In such a way we
get a family of systems depending on four parameters $b$, $e$,
$f$ and~$h$.
{\it Subcase $\kappa=0$.} Then from \eqref{val:kappa} for systems
\eqref{CS:Inf-S1-0} with $g=0$ we have $h(h-1)=0$. Without loss
of generality we can assume $h=0$ (if $h=1$ we can apply the
linear transformation which sends the straight line $ x=0$
to $y=x$). Thus we get the systems
\begin{equation}\label{CS:Inf-S1-2}
\dot x=a +cx+dy - xy,\quad \dot y=b +ex+fy -xy,
\end{equation}
for which $\mu_0=\mu_1=0$ and\ $ \mu_2=(c-e)(f-d)xy$.
We separate the proof of this subcase in three pieces.
\noindent First $\mu_2\ne0$. Then these systems possess exactly
one real singular point of multiplicity $2$ and translating it to the origin of coordinates we obtain the systems
\begin{equation}\label{CS:Inf-S1-2a}
\dot x=cx+dy - xy,\quad \dot y=ex+fy -xy,
\end{equation}
for which the condition $cf-de=0$ must be forced in order to have a double
point. Since $c^2+e^2\ne0$ (otherwise we get degenerate systems)
without loss of generality we may assume $f=qe$ and $d=qc$. In
such of way we get the family of systems
\begin{equation}\label{CS:Inf-S1-2b}
\dot x= cx+qc y - xy,\quad \dot y= ex+qey -xy,
\end{equation}
which depends on three parameters $c$, $e$ and $q\ne0$.
\noindent Second $\mu_2=0$ and $\mu_3\ne0$.
Then the singular point $M_0(0,0)$
of systems \eqref{CS:Inf-S1-2a} is simple and we must force the
condition $\mu_2=(c-e)(f-d)xy=0$. This yields either $c=e$ or
$f=d$, and in each of these cases we get a family of systems
depending on three free parameters.
\noindent Third $\mu_2=\mu_3=0$ and $\mu_4\ne0$. Since there are
no finite singularities, we shall consider the systems
\eqref{CS:Inf-S1-2} for which the condition $\mu_2=0$ yields
$(c-e)(f-d)=0$ and without loss of generality we can consider
$e=c$ via the replacing $x$ with $y$, $c$ with $f$, $d$ with $e$,
and $a$ with $b$, which keeps the form of these systems. Then we
get the systems
\[
\dot x=a +cx+dy - xy,\quad \dot y=b +cx+fy -xy,
\]
for which $\mu_0=\mu_1=\mu_2=0$ and $\mu_3=(d - f)(a-b+cd-cf)xy^2$.
Therefore the condition $\mu_3=0$ yields either $f=d$ or $a=
b-cd+cf$ and in each of these cases we get a family of systems
depending on four free parameters.
\subsection{The case of one real and two complex infinite singular points}
In this case the polynomial $C_2=yP(x,y)-xQ(x,y)$ has one
real and two complex linear factors. Therefore this cubic binary
form can be written as $C_2=x(x^2+y^2)$ via an admissible linear transformation (depending only on the coordinates of the infinite singularities of
the considered family of systems \eqref{sys:gen}; see Remark
\ref{rem:admissible-trans}). Using the
factorization \eqref{factoriz:C2} and a time rescaling we get
the family of systems
\begin{equation}\label{CS:Inf-S2}
\dot x= a+cx+dy+gx^2+(h+1)xy,\quad
\dot y=b+ex+fy-x^2+gxy+hy^2,
\end{equation}
with $\mu_0=-h\big[g^2+(h+1)^2\big]$.
\subsubsection{The case $\mu_0\ne0$} By Lemma \ref{lem:mu_i-Infty}
the unique finite singularity must be of the multiplicity four and
translating it to the origin of coordinates we get the family of systems
\begin{equation}\label{CS:Inf-SS2}
\dot x= cx+dy+gx^2+(h+1)xy,\quad
\dot y= ex+fy-x^2+gxy+hy^2.
\end{equation}
Clearly in order that the singular point $(0,0)$ has at
least of multiplicity 2 the condition $cf-de=0$ must hold.
\noindent \textit{Subcase $d\ne0$}. Then $e=cf/d$ and for systems
\eqref{CS:Inf-SS2} we obtain $\mu_4= \mu_3=0$ and
$\mu_2= -\mathcal{F}_3\big[(fg+d)x^2+ (f-dg+fh)xy- dhy^2\big]/d$,
where $\mathcal{F}_3=f(c-dg+ch)-(d^2+cdg-c^2h)$.
By Lemma \ref{lem:mu_i-(0,0)} in order to have a point of multiplicity 4
we must force the conditions $\mu_2=\mu_1=0$ to be fulfilled.
Since $d\ne0$ and $\mu_0\ne0$ (i.e. $h\ne0$) the condition $\mu_2=0$
is equivalent to $ \mathcal{F}_3=0$.
We observe that $c-dg+ch\ne0$, otherwise $g=c(h+1)/d$ and then
$ \mathcal{F}_3=-(c^2+d^2)\ne0$. So $c-dg+ch\ne0$ and from
$ \mathcal{F}_3=0$ we get
$f=(d^2+cdg-c^2h)/(c-dg+ch)$. Then we get
$\mu_1= \mathcal{F}_4\big[(d+dg^2+dh-cgh)x+h(dg-c-ch)y\big]/(d(dg-c-ch))$,
where $\mathcal{F}_4=(dg-ch)^2+h(c^2+d^2)+d^2.$\ Since $h(dg-c-ch)\ne0$
the condition $\mu_1=0$ is equivalent to $ \mathcal{F}_4=0$. We observe that
$\mathop{\rm Discrim}[\mathcal{F}_4,c]=-4d^2h\big[g^2+(h+1)^2]=4d^2\mu_0$. So for the
existence of real parameters $c,d,h$ and $g$ in order to have a point of
multiplicity 4 for a non--homogeneous quadratic system \eqref{sys:gen}
it is necessary $\mu_0>0$. Then we have either
$ c_{1,2}= (dgh\pm d\sqrt{-h\big[g^2+(h+1)^2]})/(h(h+1))= d\tilde c_{1,2}$
if $h\ne-1$, or $c_3=-dg/2=d\tilde c_3$ if $h=-1$, and this leads to the
families of systems
\begin{gather*}
\dot x= dc_0x+dy+gx^2+(h+1)xy,\\
\dot y= \frac{ dc_0 (1+c_0g-c_0^2h)}{c_0-g+c_0h}x
+\frac{ d (1+c_0g-c_0^2h)}{c_0-g+c_0h}y-x^2+gxy+hy^2,\\
\end{gather*}
where $c_0$ is one of the values $\tilde c_i$ for $i=1,2,3$. Thus in
the generic case fixing the coordinates of singularities and
fixing the multiplicity 4 for the finite singularity we obtain a
family of systems depending on three parameters $d$, $g$ and
$h$.
\noindent \textit{Subcase $d=0$}. Then we have $cf=0$ and we claim that
in order to have the singular point $M_0(0,0)$ of multiplicity 4 it
is necessary $c=0$. Indeed if $c\ne0$ then $f=0$ and for
systems \eqref{CS:Inf-SS2} with $d=f=0$ we get $\mu_4=\mu_3=0$
and $\mu_2=ch\big[-(c+eg)x^2+ (-e+cg-eh)xy+chy^2\big]$.
Since $\mu_0\ne0$ (i.e. $h\ne0$) we obtain $ch\ne0$ and this
yields $\mu_2\ne0$.
Since $c=0$ for systems \eqref{CS:Inf-SS2} we calculate $\mu_4=\mu_3=0$
and $\mu_2=f(-e+fg-eh) \big[gx +(h+1)y\big]x$.
In this case the condition $\mu_2=0$ implies $f(-e+fg-eh)=0$.
If $h\ne-1$ then calculations yield either $\mu_1=-eh(1+h)\big[gx
+(h+1)y\big]$ if $f=0$, or $\mu_1=-f[g^2+(h+1)^2]x$ if
$e=fg/(h+1)$. Since $\mu_0\ne0$ in both cases we get $e=f=0$ and
this leads to a family of homogeneous quadratic systems depending
on two parameters.
Assume now $h=-1$. Then the condition $\mu_0\ne0$ implies $g\ne0$.
Hence the conditions $\mu_2=f^2g^2x^2=0$ and $\mu_1=-2fg^2x =0$
yield $f=0$. In this case we obtain the family of systems
$$
\dot x= gx^2,\quad
\dot y= ex -x^2+gxy-y^2,
$$
which also depends on two parameters.
\subsubsection{Case $\mu_0=0$, $\mu_1\ne0$} In this case by Lemma
\ref{lem:mu_i-Infty} a single singularity of systems
\eqref{CS:Inf-S2} must be of multiplicity 3. We observe that for
systems \eqref{CS:Inf-S2} the polynomial $\mu_1$ can be
represented in the form $\mu_1=gW_1(x,y)+(h+1)W_2(x,y)$, where
$W_1$ and $W_2$ are polynomials in the coefficients of systems
\eqref{CS:Inf-S2} as well as homogeneous of degree one in $x$ and
$y$. Then we conclude that the conditions $\mu_0=0$ and
$\mu_1\ne0$ yield $h=0$ and after a translation we get the systems
\begin{equation}\label{CS:Inf-SS2Triple}
\dot x= cx+dy + gx^2+xy,\quad
\dot y= ex+fy -x^2 + gxy,
\end{equation}
for which $\mu_1=(dg-f)(g^2+1)x\ne0$. Since $(0,0)$ is of
multiplicity 3 the condition $cf-de=0$ must hold.
\noindent \textit{Subcase $d\ne0$}. Then $e=cf/d$ and for systems
\eqref{CS:Inf-SS2Triple} we obtain $\mu_4=\mu_3=0$ and
$\mu_2= \big[c(dg-f)+d(d+fg)\big]\big[(fg+d)x +(f-dg)y\big]x/d$.
By Lemma \ref{lem:mu_i-(0,0)} in order to have a point of
multiplicity 3 we must force the condition $\mu_2=0$ to be
fulfilled. Since $\mu_1\ne0$ (i.e. $dg-f\ne0$)
the condition $\mu_2=0$ is equivalent to $ c=d(d+fg)/(f-dg)$ and then
systems \eqref{CS:Inf-SS2Triple} become a family of systems which
depends on three parameters $d$, $f$ and $g$.
\noindent \textit{Subcase $d=0$}. Then we have $cf=0$ and
$\mu_1=-f(g^2+1)x\ne0$. Therefore $c=0$ and calculations yield
$\mu_4=\mu_3=0$, $\mu_2=f(fg-e) (gx+ y)x$ and $\mu_1=-f(g^2+1)x$.
Since $\mu_1\ne0$ the condition $\mu_2=0$ gives $e=fg$. So in this
case we get a family of systems which depends on two parameters
$f$ and $g$.
\subsubsection{Case $\mu_0=\mu_1=0$}
In this case according with Lemma \ref{lem:mu_i-Infty} at least
two finite singularities have gone to infinity. For systems
\eqref{CS:Inf-S2} we calculate
\begin{equation}\label{val:S2}
\mu_0=-h\big[(h+1)^2+g^2\big],\quad \kappa
=-16\big[g^2+(1+h)(1-3h)\big]
\end{equation}
and for forcing $\mu_0=0$ we have to distinguish two possibilities:
$\kappa\ne0$ and $\kappa=0$.
{\it Subcase $\kappa\ne0$.} In this case
the condition $\mu_0=0$ yields $h=0$ and then the condition
$\mu_1=(dg-f)(g^2+1)x=0$ yields $f=dg$. So we get the family of
systems
\begin{equation}\label{CS:Inf-S2-1}
\dot x=a +cx+dy + gx^2 + xy,\quad \dot y=b+ex+ dgy -x^2+gxy,
\end{equation}
with the configuration ${2\choose 1}\nu_1+{0\choose
1}\nu_2^{\,c}+{0\choose 1}\nu_3^{\,c}$ at infinity (see Lemma
\ref{lem:inf-config}) and
\begin{equation}\label{val: mu2-S2-1}
\mu_2=(ag-b+d^2+de-cdg+d^2g^2)(g^2+1)x^2.
\end{equation}
We divide the proof of this subcase in three steps.
\subparagraph{First $\mu_2\ne0$.} Then according with
Lemma \ref{lem:mu_i-Infty} the unique finite singularity must be
of multiplicity 2. So translating this singularity to the
origin we obtain the family \eqref{CS:Inf-S2-1} with $a=b=0$ and
\begin{equation}\label{val:mu2,mu3-S2}
\mu_2=d(d+e-cg+dg^2)(g^2+1)x^2,\quad \mu_3=d (e-cg)
\big[(c+eg)x+(d+e-cg+dg^2)y\big] x^2.
\end{equation}
Since $\mu_2\ne0$ the condition $\mu_3=0$ yields $e=cg$ and we
obtain the family of systems
$$
\dot x= cx+dy + gx^2 + xy,\quad \dot y=cg x+dgy -x^2+gxy,
$$
which has all singularities fixed (its configuration at infinity
corresponds to ${2\choose 1}\nu_1+{0\choose 1}\nu_2^{\,c}+{0\choose 1}\nu_3^{\,c}$ and depends on three parameters $c$, $d$ and $g$.
\subparagraph{Second $\mu_2=0$ and $\mu_3\ne0$.} Then systems \eqref{CS:Inf-S2} possess exactly one real singular point. Hence without loss of generality we can consider $a=b=0$ translating this point to the origin and we obtain systems \eqref{CS:Inf-S2-1} with $a=b=0$. For these systems we have $\mu_0=\mu_1=0$ and the values of $\mu_2$ and $\mu_3$ are given in \eqref{val:mu2,mu3-S2}. But
in this case the conditions $\mu_2=0$ and $\mu_3\ne0$ must hold.
Evidently this implies $e =cg-d-dg^2$ and we get the family of systems
$$
\dot x= cx+dy + gx^2 + xy,\quad \dot y= (cg-d-dg^2)x+ dgy -x^2+gxy,
$$
depending on three parameters $c$, $d$ and $g$.
\subparagraph{Third $\mu_2=\mu_3=0$ and $\mu_4\ne0$.} Since there are
no finite singularities we consider systems \eqref{CS:Inf-S2-1} with $\mu_0=\mu_1=0$ and the value of the polynomial $\mu_2$ is given by \eqref{val: mu2-S2-1}. Hence the condition $\mu_2=0$ yields $b=ag+d^2+de-cdg+d^2g^2 $ and then for these systems we obtain $\mu_3=(g^2+1) (a-cd+d^2g) (2d+e-cg+2dg^2)x^3$ and $\mu_4=(a-cd+d^2g)x^3W_3(a,c,d,e,g,x,y)$. Thus the conditions $\mu_3=0$ and $\mu_4\ne0$ imply $e=cg-2d(g^2+1)$. In such a way we get a family of systems which depends on four parameters $a$, $c$, $d$ and $g$.
{\it Subcase $\kappa=0$.} Then considering \eqref{val:S2} for systems
\eqref{CS:Inf-S2} the conditions $\mu_0=\kappa=~0$ yield $g=0=h+1$.
Thus we get the systems
\begin{equation}\label{CS:Inf-S2-2}
\dot x=a + cx+ dy,\quad \dot y=b+ex+fy -x^2-y^2,
\end{equation}
for which $\mu_0=\mu_1=0$ and $\mu_2=(c^2+d^2)(x^2+y^2)$.
We separate the proof in two parts.
\subparagraph{First $\mu_2\ne0$.} Now systems \eqref{CS:Inf-S2-2} have exactly one (double) real singular point. Then without loss of generality we can consider $a=b=0$ translating it to the origin of coordinates. Then we have
$\mu_4=0$ and $\mu_3=(cf-de)(cx+dy)(x^2+y^2)$,
and by Lemma \ref{lem:mu_i-(0,0)} for having a double singular point
$M_0(0,0)$ of systems \eqref{CS:Inf-S2-2} with $a=b=0$ we have to
force the condition $\mu_3=0$. Hence since $\mu_2\ne0$ we get
$cf-de=0$. Due to the fact taht $c^2+e^2\ne0$ without loss of generality we may assume $f=qe$ and $d=qc$. In such of way we get the family of systems
$$
\dot x= cx+qc y,\quad \dot y= ex+qey -x^2-y^2,
$$
which depends on three parameters $c$, $q$ and $e$.
\subparagraph{Second $\mu_2=0$.} Then we have
$\mu_2=(c^2+d^2)(x^2+y^2)=0$ which implies $c=d=0$. So we obtain
the family of systems
$$
\dot x=a,\quad \dot y=b+ex+fy -x^2-y^2,
$$
for which\ $ \mu_0=\mu_1=\mu_2=\mu_3=0,\quad
\mu_4=a^2(x^2+y^2)^2\ne0$. We observe that this family depends on
four parameters $a$, $b$, $e$ and $f$.
\subsection{The case of one double and one simple real
infinite singular points}
In this case the polynomial $C_2=yP(x,y)-xQ(x,y)$ due to a
linear transformation can be written in the form $C_2=x^2y$. Then
using the factorization \eqref{factoriz:C2} and a time rescaling
we get the family of systems
\[
\dot x= a+cx+dy+gx^2+hxy,\quad
\dot y=b+ex+fy+(g-1)xy+hy^2,
\]
for which $\mu_0=gh^2$ and $\kappa=-16h^2$.
\subsubsection{The case $\mu_0\ne0$} According with Lemma \ref{lem:mu_i-Infty}
the unique finite singularity must be of multiplicity $4$. Translating this point to the origin of coordinates we get the family of systems
\begin{equation}\label{CS:Inf-SS3}
\dot x= cx+dy+gx^2+hxy,\quad
\dot y= ex+fy+(g-1)xy+hy^2.\\
\end{equation}
Clearly in order that the singular point $(0,0)$ has at least of multiplicity 2 the condition $cf-de=0$ must hold.
\noindent \textit{Subcase $d\ne0$}. Then $e=cf/d$ and for systems
\eqref{CS:Inf-SS3} calculations yield $\mu_4=\mu_3=0$ and $\mu_2= \mathcal{F}_5\big[fgx^2+(d-dg+fh)xy-dhy^2\big]/d$, where $\mathcal{F}_5=f(dg-ch)-c(d-dg+ch)$. According with Lemma \ref{lem:mu_i-(0,0)} for having a point of multiplicity 4 we must force the conditions $\mu_2=\mu_1=0$ to be fulfilled. Since $\mu_0\ne0$ (i.e. $h\ne0$) the condition $\mu_2=0$ is equivalent to $ \mathcal{F}_5=0$.
We claim that for $dg-ch=0$ we cannot have a point of multiplicity 4. Indeed,
supposing $g=ch/d$ we get the contradiction $\mu_0=ch^3/d\ne0$ and $\mathcal{F}_5=cd=0$. So $dg-ch\ne0$ and the condition $ \mathcal{F}_5=0$ gives
$f=c(d-dg+ch)/(dg-ch)$. Then we get $\mu_1=\mathcal{F}_6\big[g(dg-d-ch)x+h(dg-ch)y\big]/({d(dg-ch)})$,
where $\mathcal{F}_6=(dg-ch)^2- d^2g$. Since $h(dg-ch)\ne0$ the
condition $\mu_1=0$ is equivalent to $ \mathcal{F}_6=0$. We observe that
$\mathop{\rm Discrim}[\mathcal{F}_6,c]=4d^2gh^2=4d^2\mu_0$. So for the existence of real
parameters $c,h,d$ and $g$ such that a non--homogeneous quadratic system \eqref{CS:Inf-SS3} has a point of multiplicity 4 it is necessary that
$\mu_0>0$. Then we have $c_{1,2}= (dg\pm d\sqrt{g })/h= d\tilde c_{1,2}$,
and this leads to the two families of systems
\begin{gather*}
\dot x= dc_0x+dy+gx^2+hxy,\\
\dot y= \frac{ d c_0^2(1- g+c_0 h)}{g-c_0h}x+
\frac{ d c_0(1- g+c_0 h)}{g-c_0h}y+(g-1)xy+hy^2,
\end{gather*}
where $c_0$ is one of the values $\tilde c_{1,2}$. Thus we get two
families of systems each of them depending on three parameters
$d$, $g$ and $h$. However fixing the coordinates of singularities
and fixing the multiplicity 4 for the finite singularity we
automatically get only one of these families depending on three
parameters.
\noindent \textit{Subcase $d=0$}. Then we have $cf=0$ and we claim that to
have the singular point $M_0(0,0)$ of multiplicity 4 it is necessary that
$c=0$. Indeed, if $c\ne0$ then $f=0$ and for systems \eqref{CS:Inf-SS3} we get
$\mu_4=\mu_3=0$ and $\mu_2=ch\big[-egx^2+(cg-c-eh)xy+chy^2\big]$.
Since $\mu_0\ne0$ (i.e. $h\ne0$) we obtain $ch\ne0$ and this yields
$\mu_2\ne0$. Thus $c=0$ and for systems \eqref{CS:Inf-SS3} obtain
$\mu_4=\mu_3=0$ and $\mu_2=f(fg-eh) (gx +hy)x$.
In this case the condition $\mu_2=0$ implies $f(fg-eh))=0$. Then
calculations yield either $\mu_1=-eh^2(gx +hy)$ if $f=0$, or
$\mu_1=fghx$ if $e=fg/h$. Since $\mu_0\ne0$ in both cases we get
$e=f=0$ and this leads to the family of homogeneous quadratic
systems depending on two parameters.
\subsubsection{The case $\mu_0=0$} Since $\kappa=-16h^2$ we shall
consider two subcases $\kappa\ne0$ and $\kappa=0$.
{\it Subcase $\kappa\neq0$.} Then $h\ne0$ and the condition $\mu_0=0$ yields $g=0$. So we get the family of systems
\begin{equation}\label{CS:Inf-S3-1}
\dot x= a+cx+dy+hxy, \quad
\dot y=b+ex+fy-xy+hy^2,\\
\end{equation}
for which $\mu_1=-h^2(c+eh)y$.
We separate the proof of this subcase in four steps.
\subparagraph{First $\mu_1\ne0$.} In this case by Lemma \ref{lem:mu_i-Infty} a single singularity of
systems \eqref{CS:Inf-S2} must be of multiplicity 3. After the
respective translation we get the systems
\begin{equation}\label{CS:Inf-SS3Triple}
\dot x= cx+dy+hxy,\quad
\dot y=ex+fy-xy+hy^2,
\end{equation}
for which $cf-de=0$ because $(0,0)$ is of multiplicity 3.
\noindent \textit{Subcase $d\ne0$}. Then $e=cf/d$ and for systems
\eqref{CS:Inf-SS3Triple} we obtain $\mu_4=\mu_3=0$,
$\mu_2= -c(d+ch+fh)\big[(fh+d)x -dhy\big]y/d$ and $\mu_1=-ch^2(fh+d)y/d$.
By Lemma \ref{lem:mu_i-(0,0)} for having a singular point of multiplicity 3
the condition $\mu_2=0$ must hold. Since $\mu_1\ne0$ the condition
$\mu_2=0$ is equivalent to $ c=-(fh+d)/h$. Thus the family of
systems \eqref{CS:Inf-SS3Triple} becomes a family which depends
on three parameters $d$, $f$ and $h$.
\noindent \textit{Subcase $d=0$}.
Then we have $cf=0$ and we claim that in order to have the singular
point $M_0(0,0)$ of multiplicity 3 it is necessary that $c=0$.
Indeed, if $c\ne0$ then $f=0$ and for systems
\eqref{CS:Inf-SS3Triple} we get $\mu_4=\mu_3=0$ and
$\mu_2=ch\big[-(eh+c)x+ chy\big]y$. Since $\kappa\ne0$ (i.e. $h\ne0$)
we obtain $ch\ne0$ and this
yields $\mu_2\ne0$, but on the other hand we must have $\mu_2=0$.
This contradiction proves our claim. Thus $c=0$ and in this case
we obtain $\mu_1=-eh^3y$ and $\mu_2=-efh^2xy$, and since
$\mu_1\ne0$ the condition $\mu_2=0$ gives $f=0$.
So we get the family of systems depending on two parameters $e$ and $h$.
\subparagraph{Second $\mu_1=0$ and $\mu_2\ne0$.} In this case by Lemma
\ref{lem:mu_i-Infty} the finite singularity must be double. Since
$\kappa\ne0$ (i.e. $h\ne0$) the condition $\mu_1=-h^2(c+eh)y=0$
yields $c=-eh$. Thus after the respective translation systems
\eqref{CS:Inf-S3-1} become the family of systems
\begin{equation}\label{CS:Inf-S3-2}
\dot x= -ehx +dy+ hxy,\quad \dot y= ex +fy -xy+ hy^2,
\end{equation}
for which we have
\begin{equation}\label{val:S3-2}
\mu_2=eh^2(d+fh+eh^2)y^2,\quad
\mu_3=e(d+fh)\big[(d+fh+eh^2)x-dhy\big]y^2
\end{equation}
and according with Lemma \ref{lem:mu_i-(0,0)} for having a double point
at the origin we must force $\mu_3=0$. Since $\mu_2\ne0$ the
condition $\mu_3=0$ yields $d=-fh$. Thus the family of systems
\eqref{CS:Inf-S3-2} becomes a family which depends on three
parameters $e$, $f$ and $h$.
\subparagraph{Third $\mu_1=\mu_2=0$ and $\mu_3\ne0$.} Then
systems \eqref{CS:Inf-S3-1} possess exactly one real singular point and due
to a translation, by the same reasons as above we get the family
of systems \eqref{CS:Inf-S3-2} and we must force $\mu_2=0$.
Considering \eqref{val:S3-2} and $\mu_3\ne0$ we have $d=-h(f +eh)$
and we again arrive to the family depending on three parameters
$e$, $f$ and $h$.
\subparagraph{Fourth $\mu_1=\mu_2=\mu_3=0$ and $\mu_4\ne0$.}
Since there are no finite singularities, we consider systems
\eqref{CS:Inf-S3-1} for which $\mu_0=0$ and due to the fact that $\kappa\ne0$ the condition $\mu_1=-h^2(c+eh)y=0$ yields $c=-eh$. Thus we obtain the
family of systems
\begin{equation}\label{CS:Inf-S3-3}
\dot x= a-ehx +dy+ hxy,\quad \dot y=b+ ex +fy -xy+ hy^2,
\end{equation}
for which the condition $\mu_2=h^2(a+de+bh+efh+e^2h^2)y^2=0$
implies $a=-(de+bh+efh+e^2h^2)$. Then calculations yield
$\mu_3=-h^2(b+ef+e^2h)(d+fh+2eh^2)y^3$ and $\mu_4=(b+ef+e^2h)y^3\,
W_4(b,d,e,f,h,x,y)$, where $W_4(b,d,e,f,h,x,y)$ is a polynomial in the indicated parameters and is linear in $x$ and $y$. Therefore the conditions
$\mu_3=0$ and $\mu_4\ne0$ give $d=-(fh+2eh^2)$ and in such a way the
family of systems \eqref{CS:Inf-S3-3} becomes a family depending
on four parameters $b$, $e$, $f$, $h$.
{\it Subcase $\kappa=0$.} Then $h=0$ and we get systems
\begin{equation}\label{CS:Inf-S3-4}
\dot x= a+cx+dy +gx^2,\quad
\dot y= b+ex+fy +(g-1)xy ,
\end{equation}
for which $\mu_0=0$ and $\mu_1=dg(g-1)^2x$.
We divide the proof of this subcase in two steps.
\subparagraph{First $\mu_1\ne0$.} In this case by Lemma \ref{lem:mu_i-Infty}
a single singularity of systems \eqref{CS:Inf-S3-4} must be of
multiplicity 3. After the respective translation we get the
systems
\begin{equation}\label{CS:Inf-SS3Triple-1}
\dot x= cx+dy +gx^2,\quad
\dot y= ex+fy +(g-1)xy ,
\end{equation}
for which $cf-de=0$ because $(0,0)$ is of multiplicity 3. Since
$\mu_1\ne0$ yields $d\ne0$. We obtain $e=cf/d$ and for systems
\eqref{CS:Inf-SS3Triple-1} we get $\mu_4=\mu_3=0$, $\mu_2= (cg-c+fg)\big[fg x +d(1-g)y\big]x$ and $\mu_1=dg(g-1)^2x$, and since $\mu_1\ne0$ the condition $\mu_2=0$ yields $f=c(1-g)/g$. Thus the family \eqref{CS:Inf-SS3Triple-1} becomes a family depending on three parameters $c$, $d$ and $g$.
\subparagraph{Second $\mu_1=0$.} According with Lemma
\ref{lem:inf-config} for $\kappa=0$ and $\mu_0=\mu_1=0$ the
configurations of infinite singularities are governed by the invariant
polynomial $L(a,x,y)$. For systems \eqref{CS:Inf-S3-4} we have
$\mu_1=dg(g-1)^2x$ and $L=8gx^2$.
\textit{1) The case $L\ne0$.} Then $g\ne0$ and the condition
$\mu_1=0$ implies $d(g-1)=0$.
${\mathbf \alpha)}$ Assume that $\mu_2\ne0$, i.e. the finite
singularity is of multiplicity 2. Then without loss of
generality translating this singular point at the origin we can
consider $a=b=0$ for systems \eqref{CS:Inf-S3-4}. Thus we obtain the systems
\begin{equation}\label{CS:Inf-S3-5}
\dot x=cx +dy+ gx^2,\quad \dot y=ex +fy +(g-1)xy,
\end{equation}
for which $d(g-1)=0$. Therefore we get
\begin{equation}\label{val:S3-5}
\mu_2=fg(fg+c-cg)x^2,\quad \mu_3=(de-cf)\big[egx+(fg+c-cg)y\big]x.
\end{equation}
Since $\mu_2\ne0$ the condition $\mu_3=0 $ yields $de-cf=0$. As
$f\ne0$ without loss of generality we introduce a new parameter $q$ through
$e=qf$, and then we get $c=dq$. Thus we obtain the family of
systems
$$
\dot x=dq x +dy+ gx^2,\quad \dot y=fqx +fy +(g-1)xy,
$$
for which $d(g-1)=0$. Therefore for $d=0$ (respectively $g=1$)
we get a family of systems depending on three parameters $f$, $q$
and $g$ (respectively $f$, $q$ and $d$).
${\mathbf \beta)}$ For $\mu_2=0$ and $\mu_3\ne0$ systems
\eqref{CS:Inf-S3-4} possess exactly one simple real singular point
and doing a translation, by the same reasons as above,
we get the family of systems \eqref{CS:Inf-S3-5} and we must force
$\mu_2=0$, taking into account the relation $d(g-1)=0$.
If $d=0$ from \eqref{val:S3-5} and since $\mu_3\ne0$ we have $f\ne0$ and
then the conditions $\mu_2=0 $ implies $f=c(g-1)/g$. Thus we
get the family of systems
$$
\dot x=cx + gx^2,\quad \dot y=ex +c(g-1)y/g +(g-1)xy,
$$
depending on three parameters $c$, $e$ and $g$.
Assume $d\ne0$. Then $g=1$ and from \eqref{val:S3-5} the
condition $\mu_2=f^2x^2=0$ yields $f=0$. So we again get
the family of systems
$$
\dot x=cx + dy +x^2,\quad \dot y=ex,
$$
depending on three parameters $c$, $d$ and $e$.
${\mathbf \gamma)}$ Assume finally that $\mu_2=\mu_3=0$ and $\mu_4\ne0$,
i.e. systems \eqref{CS:Inf-S3-4} have no finite singularities.
Then considering the condition $L\ne0$
(i.e. $g\ne0$) we obtain as above that the condition
$\mu_1=dg(g-1)^2x=0$ yields $d(g-1)=0$.
Suppose $d=0$. Then we calculate
$\mu_2=g\big[a(g-1)^2-cf(g-1)+f^2g\big]x^2$.
If $g\ne1$ then since $g\ne0$ the condition
$\mu_2=0$ gives $a=[cf(g-1)-f^2g]/(g-1)^2$ and we obtain
$\mu_3=g(b+ef-bg)(c-cg+2fg)x^3/(1-g)$ and
$\mu_4=(b+ef-bg)x^3W_5(b,c,e,f,g,x,y)/(1-g)^2$,
where $W_5(b,c,e,f,g,x,y)$ is a polynomial in the indicated parameters
and linear in $x$ and $y$. Thus the conditions $\mu_3=0$ and
$\mu_4\ne0$ imply $c=2fg/(g-1)$. So we obtain the family of
systems
$$
\dot x=g(f-x+gx)^2/(g-1)^2,\quad
\dot y=b +ex+fy+(g-1)xy
$$
depending on four parameters $b$, $e$, $f$ and $g$.
If $g=1$ then $\mu_2=f^2x^2=0$ gives $f=0$ and then $\mu_3=0$.
This leads to the family
$$
\dot x=a+cx+x^2,\quad
\dot y=b +ex,
$$
which also depends on four parameters $a$, $b$, $c$ and $e$.
Assume now $d\ne0$. Hence the condition $\mu_1=0$ gives $g=1$ and
from $\mu_2=f^2x^2=0$ we get $f=0$. Then $\mu_3=de^2x^3=0$ and since
$d\ne0$ we obtain $e=0$. Thus we get the family of systems
$$
\dot x=a+cx+dy+x^2,\quad
\dot y=b
$$
depending on four parameters $a$, $b$, $c$ and $d$.
\textit{2) The case $L=0$.} Then $g=0$ and systems
\eqref{CS:Inf-S3-4} become
\[
\dot x= a+cx+dy,\quad \dot y= b+ex+f xy,
\]
for which we have $\mu_1=0$ and $\mu_2=-cd\,xy$.
${\mathbf \alpha)}$ Assume that $\mu_2\ne0$, i.e. the finite
singularity is of multiplicity 2. Then translating this singular
point at the origin we can consider $a=b=0$. Thus we get the systems
\begin{equation}\label{CS:Inf-S3-6}
\dot x=cx +dy,\quad \dot y=ex +fy -xy,
\end{equation}
for which calculations yield $\mu_2=-cd\,xy$ and $\mu_3=(de-cf)(cx+dy)xy$.
Since $\mu_2\ne0$ (i.e. $cd\ne0$) the condition $\mu_3=0$ gives
$e=cf/d$ and hence the family of systems \eqref{CS:Inf-S3-6}
becomes a family depending on three parameters $c$, $d$ and $f$.
${\mathbf \beta)}$ Suppose that $\mu_2=0$ and $\mu_3\ne0$.
By Lemma \ref{lem:mu_i-Infty} there exists exactly one
simple real singular point on the phase plane of systems
\eqref{CS:Inf-S3-4} and doing a translation we
get the family of systems \eqref{CS:Inf-S3-6} for which we must
force $\mu_2=-cd\,xy=0$. So $cd=0$ and since $\mu_3\ne0$ the
condition $c^2+d^2\ne0$ holds. We conclude that in the case $c=0$
as well as in the case $d=0$ the family of systems
\eqref{CS:Inf-S3-6} becomes a family depending respectively on
the remaining three parameters.
${\mathbf \gamma)}$ Assume finally $\mu_2=\mu_3=0$ and $\mu_4\ne0$,
i.e. systems \eqref{CS:Inf-S3-4} have no finite singularities.
For these systems we have $\mu_2=-cd\,xy$ and the condition
$\mu_2=0$ yields $cd=0$. By Lemma \ref{lem:inf-config}
in this case the configurations of infinite singularities are
governed by the invariant polynomial $\kappa_1=-32d$.
If $\kappa_1\ne0$ (i.e. $d\ne0$) then $c=0$ and we calculate
$\mu_3=d(a+de)x^2y$. Hence $\mu_3=0$ gives $a=-de$ and we get the
family of systems
$$
\dot x=d(y-e),\quad
\dot y=b+ex+fy-xy,
$$
depending on four parameters $b$, $d$, $e$ and $f$. The
configuration of infinite singular points corresponds to
${3\choose 1}p+{1\choose 2}q$.
Assume $\kappa_1=0$. Then $d=0$ and for systems
\eqref{CS:Inf-S3-4} we obtain $\mu_3=-c(a+cf)x^2y$ and
$K_1=-cx^2y$. Thus for $K_1\ne0$ the condition $\mu_3=0$ gives
$a=-cf$ and we get the family of systems
$$
\dot x=c(x-f),\quad
\dot y=b+ex+fy-xy,
$$
depending on four parameters $b$, $c$, $e$, $f$ and with the
configuration ${1\choose 1}p+{3\choose 2}q$ at infinity.
If $K_1=0$ we have $c=0$ and this leads to the family of systems
$$
\dot x=a,\quad
\dot y=b+ex+fy-xy,
$$
which also depends on four parameters $a$, $b$, $e$ and $f$, but
with the configuration ${2\choose 1}p+{2\choose 2}q$ at infinity.
\subsection{Systems with one triple real infinite singular point}
In this case the polynomial $C_2=yP(x,y)-xQ(x,y)$ due to an
admissible linear transformation can be written into the form
$C_2=x^3$. Then using the factorization \eqref{factoriz:C2} as
well as a time rescaling we get the family of systems
\begin{equation}\label{CS:Inf-S4}
\dot x= a+ cx+dy+gx^2+hxy,\quad
\dot y= b+ex+fy-x^2+gxy+hy^2,\\
\end{equation}
for which $\mu_0=-h^3$.
\subsubsection{The case $\mu_0\ne0$} By Lemma \ref{lem:mu_i-Infty}
the unique finite singularity must be of multiplicity four and
translating this point to the origin of
coordinates we get the family of systems
\begin{equation}\label{CS:Inf-S4-1}
\dot x= cx+dy+gx^2+hxy,\quad
\dot y= ex+fy-x^2+gxy+hy^2.\\
\end{equation}
Clearly the singular point $(0,0)$ will be at least of multiplicity 2
if the condition $cf-de=0$ holds.
If $d\ne0$ then $e=cf/d$ and for systems \eqref{CS:Inf-S4-1} we get
$\mu_4=\mu_3=0$ and $\mu_2= \mathcal{F}_7\big[-(d+fg)x^2 +(dg-fh)xy+dhy^2\big]/d$, where $\mathcal{F}_7=(c+f)(ch-dg)-d^2$. So since
$\mu_0\ne0$ (i.e. $h\ne0$) the condition $\mu_2=0$ implies
$\mathcal{F}_7=0$. We observe that $ch-dg\ne0$, otherwise we get
the contradiction $\mathcal{F}_7= -d^2=0$. So the condition
$\mathcal{F}_7=0$ gives $f=d^2/(ch-dg)-c$ and we obtain
$\mu_1= \mathcal{F}_8\big[(dg^2+dh-cgh)x+h(dg-ch)y\big]/({d(dg-ch)})$,
where $\mathcal{F}_8=(dg-ch)^2+ d^2h$. Since
$h(dg-ch)\ne0$ the condition $\mu_1=0$ is equivalent to $
\mathcal{F}_8=0$. We observe that $\mathop{\rm Discrim}[\mathcal{F}_8,c]=-4d^2h^3= 4d^2\mu_0$. So it is necessary that $\mu_0>0$ in order that a non--homogeneous quadratic system \eqref{CS:Inf-S4-1} has a singular point of multiplicity 4. Then we have $c_{1,2}= (dg\pm d\sqrt{-h })d= d\tilde c_{1,2}$,
and this leads to the two families of systems
\begin{gather*}
\dot x= dc_0x+dy+gx^2+hxy,\\
\dot y= -\frac{ d c_0(1+c_0 g-c_0^2 h)}{g-c_0h}x
-\frac{ d (1+c_0 g-c_0^2 h)}{g-c_0h}y-x^2+gxy+hy^2,
\end{gather*}
where $c_0$ is one of the values $\tilde c_{1,2}$ above. Thus we
get two families of systems each of them depending on 3
parameters. However fixing the coordinates of singularities and
fixing the multiplicity 4 for the finite singularity we automatically
get only one of these families depending on three parameters.
Assume now $d=0$. Then we have $cf=0$ and we claim that for having
the singular point $M_0(0,0)$ of multiplicity 4 it is necessary that $c=0$.
Indeed, if $c\ne0$ then $f=0$ and for systems \eqref{CS:Inf-S4-1} we get
$\mu_4=\mu_3=0$ and $\mu_2=ch\big[-(c+eg)x^2 +(cg-eh) xy+chy^2\big]$.
Since $\mu_0\ne0$ (i.e. $h\ne0$) we obtain $ch\ne0$ and this
yields $\mu_2\ne0$. Thus $c=0$ and for systems \eqref{CS:Inf-S4-1} we get
$\mu_4=\mu_3=0$, $\mu_2=f(fg-eh)(gx+hy)x$ and $\mu_1= h \big[(fg^2-fg-egh)x+h(fg-eh)y)\big]$. In this case the conditions $\mu_2=\mu_1=0$ imply $e=f=0$ and this leads to the family of homogeneous quadratic systems
$$
\dot x= gx^2+hxy,\quad \dot y= -x^2+gxy+hy^2,
$$
which depends on two parameters.
\subsubsection{Case $\mu_0=0$ and $\mu_1\ne0$} In this case by Lemma
\ref{lem:mu_i-Infty} the single singularity of systems
\eqref{CS:Inf-S4} must be of multiplicity 3. The condition
$\mu_0=0$ yields $h=0$ and doing a translation we obtain the systems
\begin{equation}\label{CS:Inf-S4-2}
\dot x= cx+dy+gx^2,\quad
\dot y= ex+fy-x^2+gxy,
\end{equation}
for which $\mu_1=dg^3x\ne0$. Clearly the singular point $(0,0)$ will be at
least of multiplicity 2 if the condition $cf-de=0$ holds. Since
$d\ne0$ we can write $e=cf/d$, and then $\mu_4=\mu_3=0$ and $\mu_2= (d+cg +fg)\big[(fg+d)x -dgy\big]x$. Since $\mu_1\ne0$ the condition $\mu_2=0$ yields $f=-(d+cg)/g$. Thus we obtain the family
$$
\dot x= cx+dy +gx^2,\quad
\dot y= -c(cg+d)x/(dg)-(cg+d)y/g-x^2+ gxy,\\
$$
depending on three parameters $c$, $d$ and $g$.
\subsubsection{Case $\mu_0=\mu_1=0$ and $\mu_2\ne0$} By \ref{lem:mu_i-Infty} a single singularity of systems \eqref{CS:Inf-S4} must be of multiplicity 2. Therefore after a translation we obtain systems \eqref{CS:Inf-S4-2} for which $\mu_1=dg^3x=0$ and the singular point $(0,0)$ is double. So $dg=0$ and then $\mu_2=(d^2-cfg^2+f^2g^2)x^2\ne0$. Moreover we must force the condition $cf-de=0$ to have a double singular point at the origin.
If $d=0$ we have $cf=0$ and since $f\ne0$ (otherwise
systems \eqref{CS:Inf-S4-2} are degenerate) we obtain $c=0$. This
leads to the systems
$$
\dot x= gx^2,\quad \dot y= ex+fy -x^2+gxy
$$
depending on three parameters $e$, $f$ and $g$.
If $g=0$ then $\mu_2=d^2x^2\ne0$ and then we have $e=cf/d$. This
leads to the family of systems
$$
\dot x= cx+dy,\quad
\dot y= cfx/d+fy -x^2,
$$
depending also on three parameters $c$, $d$ and $f$.
\subsubsection{Case $\mu_0=\mu_1=\mu_2=0$, $\mu_3\ne0$}
Then systems \eqref{CS:Inf-S4} possess exactly one simple real
singular point and due to a translation we shall consider
systems \eqref{CS:Inf-S4-2} with the conditions $\mu_1=dg^3x=0$
and $\mu_2=(d^2-cfg^2+f^2g^2)x^2=0$.
\noindent \textit{Subcase $g=0$.} Then the condition $\mu_2=d^2x^2=0$ gives
$d=0$ and we get the family
$$
\dot x= cx,\quad
\dot y= ex+fy -x^2,\\
$$
which depends on three parameters $c$, $e$ and $f$.
\noindent \textit{Subcase $g\ne0$}. Then $d=0$ and for systems
\eqref{CS:Inf-S4-2} we obtain $\mu_2= fg^2(f-c)x^2$ and $\mu_3=-cf\big[(c+eg)x+g(f-c)y\big]x^2$,
and since $\mu_3\ne0$ and $g\ne0$ the condition $\mu_2=0$ implies
$f=c$. This leads to the family of systems
$$
\dot x= cx+gx^2,\quad
\dot y= ex+cy -x^2+gxy\\
$$
depending also on three parameters $c$, $e$ and $g$.
\subsubsection{Case $\mu_0=\mu_1=\mu_2=\mu_3=0$ and $\mu_4\ne0$} In this case systems \eqref{CS:Inf-S4} have no finite singularities and the condition
$\mu_0=0$ yields $h=0$. So we obtain the systems
\begin{equation}\label{CS:Inf-S4-3}
\dot x= a+cx+dy+gx^2,\quad
\dot y=b+ex +fy -x^2+gxy
\end{equation}
for which the condition $\mu_1=0$ yields $dg=0$.
\noindent \textit{Subcase $g=0$.} In this case for systems
\eqref{CS:Inf-S4-3} the condition $\mu_2=d^2x^2=0$ gives $d=0$
and then $\mu_3=-c^2fx^3=0$ and $\mu_4=\big[(a^2+ace-bc^2)x+acfy\big] x^3\ne0$. Thus we get the 5--parameter family of systems
$$
\dot x= a+cx,\quad \dot y=b+ex +fy -x^2
$$
for which the condition $cf=0$ holds. Clearly this condition, in
each of two cases, leads to the family depending respectively on
the remaining four parameters.
\noindent \textit{Subcase $g\ne0$.} Then $d=0$ and for systems
\eqref{CS:Inf-S4-3} the condition $\mu_2=g^2(ag-cf+f^2)x^2=0$
implies $a=f(c-f)/g$. This leads to the systems
$$
\dot x= (f+gx)(c-f+gx)/g,\quad
\dot y=b +ex+fy -x^2+gxy
$$
for which $\mu_4=(f^2+efg-bg^2)
x^3 W_6(b,c,e,f,g,x,y)/g^2$ and $\mu_3= (c-2f)(f^2+efg-bg^2)x^3$, where $W_6(b,c,e,f,g,x,y) $ is a polynomial in the given parameters and is linear in $x$ and $y$. Thus, since $\mu_4\ne0$ the
condition $\mu_3=0$ gives $c=2f$ and this leads to the family of
systems depending on four parameters $b$, $e$, $f$ and $g$.
\section{Systems with the infinite line full of singularities}
Assume that the polynomial $C_2=yP(x,y)-xQ(x,y)=0$ in $\mathbb{R}[x,y]$.
Clearly this class of quadratic systems have the form
\begin{equation}\label{CS:Inf-S5}
\dot x= a+cx+dy+gx^2+hxy,\quad \dot y=b+ex +fy +gxy+hy^2,
\end{equation}
for which $\mu_0=0$. So by Lemma \ref{lem:mu_i-Infty} these systems can possess finite singularities of total multiplicity at most three.
\pagebreak
\subsection{Systems with finite singularities of total multiplicity 3}
\subsubsection{Systems with three finite real simple singularities} Assuming that $M_i(x_i,y_i)$ (i=1,2,3) are real
distinct singular points of systems \eqref{CS:Inf-S5}, due to an
admissible affine transformation we can move them to the points
$(0,0)$, $(0,1)$ and $(1,0)$. Therefore we get the family of
systems
$$
\dot x= cx -cx^2-fxy,\quad \dot y=fy -cxy-f y^2,
$$
which up to a time rescaling depends on a single parameter.
\subsubsection{Systems with one real and two complex finite singularities}
As it was mentioned early (see Subsection \ref{Subsec:1+2c}) due
to an admissible affine transformation which moves the respective
singularities to the points $M_{1,2}(0,\pm i)$, $M_3(1,0)$ we get
the family of quadratic systems
\begin{gather*}
\dot x = a -(a+g)x + gx^2 +2 h xy + ay^2,\\
\dot y = b -(b+l)x + lx^2 +2 m xy + b y^2.
\end{gather*}
Now taking into consideration the identity $C_2(a,x,y)=0$ in
$\mathbb{R}[x,y]$ we get $a=l=~0$, $m=g/2$ and $h=b/2$. Hence we get the
family of systems
$$
\dot x= -gx + gx^2 + b xy ,\quad \dot y= b -b x + g xy +b y^2,
$$
which up to a time rescaling depends on one parameter.
\subsubsection{Systems with one simple and one double real
finite singularities}
Via an admissible affine transformation we can assume that the two
distinct singularities of systems \eqref{CS:Inf-S5} are placed at
the points $(0,0)$ and $(1,0)$. Then evidently we get the
relations $a=b=e=0$, $g=-c$ and these systems become
\begin{equation}\label{CS:Inf-S5-1}
\dot x= cx+dy-cx^2+hxy,\quad \dot y= fy -cxy+hy^2,
\end{equation}
and one of the singular points, say $(0,0)$, is double. Then the
relation $cf=0$ must hold. However the relation $c=0$ yields
degenerate systems and, hence we obtain $f=0$. Clearly this leads
to a family of systems which up to a time rescaling depends on
two parameters.
\subsubsection{Systems with one finite real triple singularity}
In this case due to a translation we have the systems
\eqref{CS:Inf-S5} with $a=b=0$ (then $\mu_4=0$) and we must force
$\mu_3=\mu_2=0$ and $\mu_1\ne0$ in order to have a point of
multiplicity 3 at the origin. Thus for systems \eqref{CS:Inf-S5}
with $a=b=0$ we have $cf-de=0$.
\noindent \textit{The case $d\ne0$.} Then we have $e=cf/d$ and calculations
yield $\mu_4=\mu_3=0$,
\begin{equation}\label{val:mu_i-C2-0}
\mu_2= \frac{c+f}{d}(d g - c h)(f x - d y) (g x + h
y),\quad \mu_1= \frac{d g - c h}{d}(d g + f h) (g x + h y).
\end{equation}
So taking into consideration Lemma \ref{lem:mu_i-Infty} we obtain
that the conditions $\mu_2=0$ and $\mu_1\ne0$ imply $f=-c$. Then
we get the following family of systems
$$
\dot x= cx+dy+gx^2+hxy,\quad \dot y= -c^2x/d-cy +gxy+hy^2,
$$
which up to a time rescaling depends on three parameters.
\noindent \textit{The case $d=0$.} In this case we obtain $cf=0$. Then for
systems \eqref{CS:Inf-S5} with $a=b=d=cf=0$ we have
$\mu_4=\mu_3=0$ and for $c=0$ (respectively, $f=0$) we calculate
$\mu_2= f(f g - e h)(g x + h y)x$ and $\mu_1= h(f g - e h)(g x + h
y)$ (respectively $ \mu_2= ch(g x + h y)(cy-ex)$ and $\mu_1= -h(cg
+e h)(g x + h y)$). As we can see in both cases the conditions $\mu_2=0$ and $\mu_1\ne0$ imply $c=f=0$ and we get the family of systems
$$
\dot x= x(gx+hy),\quad \dot y= ex +gxy+hy^2,
$$
which up to a time rescaling depends on two parameters.
\subsection{Systems with finite singularities of total multiplicity 2}
By Lemma \ref{lem:mu_i-Infty} for this class of systems
the conditions $\mu_0=\mu_1=0$ and $\mu_2\ne0$ have to be
fulfilled.
\subsubsection{Systems with two finite real simple singular
points} In this case we can consider systems
\eqref{CS:Inf-S5-1} possessing two singularities $(0,0)$ and
$(1,0)$. For systems \eqref{CS:Inf-S5-1} we obtain
$\mu_1= -c (c d + c h - f h) (c x - h y)$ and $\mu_2= -c (c x - h y)\big[f(c-f)x+ (c d + d f + c h - f h) y\big]$.
Hence the conditions $\mu_1=0$ and $\mu_2\ne0$ yield $c\ne0$ and
$d=h(f-c)/c$. This leads to the following family of systems
$$
\dot x= cx+h(f-c)y/c-cx^2+hxy,\quad \dot y= y (f - c x + h y),
$$
which up to a time rescaling depends on two parameters.
\subsubsection{Systems with two finite complex singular
points} In this case according to Lemma \ref{lem:ImSP} via an
admissible affine transformation a quadratic system can be brought
to the canonical form \eqref{CS:c1c1XX} with the singularities
$\widetilde M_{1,2}(0,\pm i)$. For these systems the identity
$C_2=0$ yields $a=l=0$, $m=g/2$ and $h=b/2$. Thus we get the
family of systems
\begin{equation}\label{CS:c1c1XX-C2=0}
\dot x= cx + gx^2 +bxy,\quad \dot y= b +ex +gxy +b y^2,
\end{equation}
for which the conditions $\mu_0=\mu_1=0$ and $\mu_2\ne0$ must
hold. For systems \eqref{CS:c1c1XX-C2=0} calculations yield
$\mu_0=0$, $\mu_1= -b (be+cg)(g x+b y)$ and
$\mu_2=b(g x+b y)\big[(bg-ce)x+ (b^2 + c^2) y\big]$.
Thus $b\ne0$ and the condition $\mu_1=0$ implies $e=-cg/b$.
Hence we get a family of systems which up to a time rescaling
depends on two parameters.
\subsubsection{Systems with one double real finite singular
point} In this case due to a translation we obtain systems
\eqref{CS:Inf-S5} with $a=b=0$ (then $\mu_4=0$) and by Lemmas
\ref{lem:mu_i-Infty} and \ref{lem:mu_i-(0,0)} we must force
$\mu_3=\mu_1=0$ and $\mu_2\ne0$ in order to have a point of
multiplicity 2 at the origin and no more finite singularities.
Thus for systems \eqref{CS:Inf-S5} with $a=b=0$ we shall set
$cf-de=0$ in order to have a multiple point at the origin.
\textit{The case $d\ne0$.} Then we have $e=cf/d$ (in this case
$\mu_3=0$) and we get the values of the polynomials $\mu_1$ and
$\mu_2$ given in the formulas \eqref{val:mu_i-C2-0}. Since
$\mu_2\ne0$ the condition $\mu_1=0$ implies $g=-f h/d$. Thus we
get the following family of systems
$$
\dot x= cx+dy-fhx^2/d+hxy,\quad \dot y= cfx/d+fy -fh xy/d+hy^2,
$$
which up to a time rescaling depends on three parameters.
\textit{The case $d=0$.} In this case the point $(0,0)$ will be a
double singular point for systems \eqref{CS:Inf-S5} with $a=b=0$ if $cf=0$.
For $c=0$ we get the family of systems
$$
\dot x= gx^2 +hxy,\quad \dot y= ex+fy +g xy +hy^2,
$$
for which $\mu_0=\mu_3=\mu_4=0$, $\mu_1= h (f g - e h) (g x + h y)$
and $\mu_2=f (f g - e h) x (g x + h y)$. Hence the conditions
$\mu_1=0$, $\mu_2\ne0$ imply $h=0$ and we get a family of systems
which up to a time rescaling depends on two parameters.
In the case $f=0$ we obtain $\mu_0=\mu_3=\mu_4=0$,
$\mu_1= -h (c g + e h) (g x + h y)$ and $\mu_2=c h (-e x + c y) (g x + h y)$.
So from $\mu_1=0$, $\mu_2\ne0$ we obtain $h\ne0$, $e=-cg/h$ and we
again have a family of systems which up to a time rescaling depends on
two parameters.
\subsection{Systems with finite singularities of total multiplicity
less than or equal to 1}
\subsubsection{Systems with one simple finite singular
point} In this case due to a translation we obtain systems
\eqref{CS:Inf-S5} with $a=b=0$ (then $\mu_4=0$) and by Lemma
\ref{lem:mu_i-Infty} we must force $\mu_1=\mu_2=0$ and $\mu_3\ne0$ in
order to have a simple singular point at the origin and no more finite
singularities. Thus for systems \eqref{CS:Inf-S5} with $a=b=0$ we
calculate
\begin{equation}\label{val:mu_1-r1}
\mu_0=\mu_4=0,\quad \mu_1= (dg^2-cgh+fgh-eh^2)(g x + h y).
\end{equation}
\textit{The case $g\ne0$.} Then $\mu_1=0$ yields
$d=h(cg-fg+eh)/g^2$ and calculations yield
\begin{gather*}
\mu_2=\frac{1}{g^2}(eh-fg)(c g - f g + 2 e h)(g x + h y)^2 ,\\
\mu_3=\frac{1}{g^4}(eh-fg)(c g + e h)(g x + h y)^2[egx+(fg-cg-eh)y].
\end{gather*}
Hence the conditions $\mu_2=0$ and $\mu_3\ne0$ imply
$f=(cg+2eh)/g$ and we get the family of systems
$$
\dot x= cx-eh^2y/g^2+gx^2+hxy,\quad \dot y=
ex+(cg+2eh)y/g+gxy+hy^2,
$$
which up to a time rescaling depends on three parameters.
\textit{The case $g=0$.} Then $ \mu_1= -eh^3 y$ and $\mu_3=h(de-cf)y[ex^2+(f-c)cxy-dy^2]$,
and since $\mu_3\ne0$ the condition $\mu_1=0$ implies $e=0$. Then
$\mu_2=c(c-f)h^2y^2$, $\mu_3=cfhy^2(cx-fx+dy)$ and hence the
conditions $\mu_2=0$ and $\mu_3\ne0$ yield $f=c$. In such a way in
this case we obtain the family of systems
$$
\dot x= cx+dy +hxy,\quad \dot y=cy+hy^2,
$$
which up to a time rescaling depends on two parameters.
\subsubsection{Systems without finite singularities}
By Lemma \ref{lem:mu_i-Infty} for systems
\eqref{CS:Inf-S5} (for which $\mu_0=0$) we must force the
conditions $\mu_1=\mu_2=\mu_3=0$ and $\mu_4\ne0$. For these systems
we have the value of $\mu_1$ indicated in formulas
\eqref{val:mu_1-r1}. Therefore following the above way we
consider two cases $g\ne0$ and $g=0$.
\textit{The case $g\ne0$.} Then $\mu_1=0$ yields
$d=h(cg-fg+eh)/g^2$ and we get $\mu_2=\big[ag^3-f(c-f)g^2+gh(ce-3ef+bg)+2e^2h^2\big](gx+hy)^2/g^2$,
and evidently the relation $\mu_2=0$ yields $a=\big[
f(c-f)g^2-gh(ce-3ef+bg)-2e^2h^2\big]/g^3$. Then we have
$\mu_3=-\frac{1}{g^4}(c g - 2 f g + 3 e h)(bg^2
-efg+e^2h)(gx+hy)^2$ and $\mu_4=(bg^2 -efg+e^2h)W_7(x,y)$,
and the conditions $\mu_3=0$ and $\mu_4\ne0$ imply
$c=(2fg-3eh)/g$. Thus we get the family of systems
\begin{gather*}
\dot x= \frac{(fg-eh)^2-bg^2h}{g^3}+ \frac{2fg-3eh}{g}\,x
+\frac{h(fg-2eh)}{g^2}\,y +gx^2+hxy,\\
\dot y=b+ex+fy +gxy+hy^2,
\end{gather*}
which up to a time rescaling depends on four parameters.
\textit{The case $g=0$.} Then $\mu_1= -eh^3 y$ and $\mu_4=h W_8(b,c,e,f,g,
h,x,y)$, and since $\mu_4\ne0$ the condition $\mu_1=0$ gives $e=0$. In this
case from $\mu_2=h^2(c^2-cf+bh)y^2\!=\!0$ we obtain $b=c(f-c)/h$ and
then we have $\mu_3= h(2c-f)(cd-ah) y^3$ and $\mu_4=(cd-ah) W_{9}(b,c,e,f,g,
h,x,y)$. Hence, since $\mu_4\ne0$ the condition $\mu_3=0$ implies $f=2c$ and
we get the family of systems
$$
\dot x= a+ cx +dy +hxy,\quad \dot y=(c+hy)^2/h,
$$
which up to a time rescaling depends on three parameters.
Since all the possible cases were examined the Main Theorem is
proved.
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