\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 119, pp. 1--23.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/119\hfil Extended Kukles system]
{Centres and limit cycles for an extended Kukles system}

\author[J. M. Hill, N. G. Lloyd, J. M. Pearson\hfil EJDE-2007/119\hfilneg]
{Joe M. Hill, Noel G. Lloyd, Jane M. Pearson}  % in alphabetical order

\address{Joe M. Hill \newline
Institute of Mathematical and Physical Sciences,
Aberystwyth University,
Ceredigion SY23 3BZ, Wales, UK}
\email{jyh@aber.ac.uk}

\address{Noel G. Lloyd \newline
Institute of Mathematical and Physical Sciences,
Aberystwyth University,
Ceredigion SY23 3BZ, Wales, UK}
\email{ngl@aber.ac.uk}

\address{Jane M. Pearson \newline
Institute of Mathematical and Physical Sciences,
Aberystwyth University,
Ceredigion SY23 3BZ, Wales, UK}
\email{jmp@aber.ac.uk}

\thanks{Submitted August 10, 2007. Published September 6, 2007.}
\subjclass[2000]{34C07, 37G15}
\keywords{Nonlinear differential equations; invariant curves; limit cycles}

\begin{abstract}
 We present conditions for the origin to be a centre for a class
 of cubic systems.  Some of the centre conditions are determined
 by finding complicated invariant functions.  We also investigate
 the coexistence of fine foci and the simultaneous bifurcation of
 limit cycles from them.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

In this paper we establish some properties of the cubic differential system
\begin{equation} \label{kukplus}
\begin{gathered}
\dot{x}  =P(x,y)=\lambda x+y+kxy,\\
\dot{y}=Q(x,y)
       =-x+\lambda y+a_{1}x^{2}+a_{2}xy+a_{3}y^{2}+a_{4}x^{3}
        +a_{5}x^{2}y+a_{6}xy^{2}+a_{7}y^{3},
\end{gathered}
\end{equation}
where the $a_{i}$ and $k$ are real. We first became
interested in this  class of systems when considering
transformations to generalised Li\'enard form \cite{waa}.  It was
also brought to our attention that a system used to model
predator-prey interactions with intratrophic predation could be
transformed so that it is an example of a system of type
\eqref{kukplus}.  We investigated this particular case of system
\eqref{kukplus} in \cite {pred-prey}. In \cite{isoc} we found
conditions for the origin to be an isochronous centre for system
\eqref{kukplus}.

When $\lambda=0$ the origin is said to be a \textit{fine focus};
then system \eqref{kukplus} is derived from a second order scalar
equation and it has an invariant line $kx=-1$. When $k=0$,
\eqref{kukplus} is often referred to as the Kukles system; this
system has been extensively studied, see \cite{cjcnglkukles},
\cite{kukles3} and \cite{wuchenyang} for example.

Here we derive conditions for the origin to be a centre for
system \eqref{kukplus} and consider the simultaneous bifurcation
of limit cycles from several fine foci. We shall see, for example,
that at most two fine foci of \eqref{kukplus} can coexist; when
one fine focus is of order one, the other is of maximum order six
and when one fine focus is of order two, the other is of maximum
order two. We show that in the latter case a large amplitude limit
cycle can surround the two fine foci and conjecture that this is
also true in the former.

We obtain necessary conditions for a critical point to be a centre
for \eqref{kukplus} by calculating the \textit{focal values},
which are polynomials in the coefficients $k,a_{i}$.  There is a
function $V$, analytic in a neighbourhood of the origin, such that
its rate of change along orbits, $\dot{V}$, is of the form
$\eta_{2}r^{2}+\eta_{4}r^{4}+\cdots$, where $r^{2}=x^{2}+y^{2}$.
The $\eta_{2j}$ are the focal values and the origin is a centre
if, and only if, they are all zero. The relations
$\eta_{2}=\eta_{4}=\cdots=\eta_{2j}=0$ are used to eliminate some
of the variables from $\eta_{2j+2}$.  This reduced focal value
$\eta_{2j+2}$, with strictly positive factors removed, is known as
the \textit{Liapunov quantity }$L(j)$. We note that
$L(0)=\lambda$.  The circumstances under which the calculated
$L(j)$ are zero yield possible centre conditions. The origin is a
fine focus of \textit{order} $j$ if $L(i)=0$ for $i=0,1,\dots,j-1$
and $L(j)\neq0$; at most $j$ small amplitude limit cycles can
bifurcate from a fine focus of order $j$.

Various methods are used to prove the sufficiency of centre
conditions; in this paper we require three of them. The simplest
is  that the origin is a centre if the system is symmetric in
either axis, that is, it remains invariant under the
transformation ($x,y,t)\mapsto(x,-y,-t)$ or
$(x,y,t)\mapsto(-x,y,-t)$. Another technique which we employ
involves a transformation of the system to Li\'{e}nard form
\begin{equation}
\dot{x}=y,\quad\dot{y}=-f(x)y-g(x).\label{lienard}
\end{equation}
The relevant results are as follows; proofs can be found
in \cite{CLPCherkas}.

\begin{lemma}\label{lemma1}
Consider system \eqref{lienard} where $f,g$ are analytic, $g(0)=0$, 
$xg(x)>0$ for $x\neq 0$ and $g'(0)>0$.  
Let $F(x)=\int_{0}^{x}f(\mu)d\mu$ and $G(x)=\int_{0}^{x}
g(\mu)d\mu$.
\begin{itemize}
\item[(i)] The origin is a centre for system (\ref{lienard}) if and only
if there is an analytic function $\Phi$ with $\Phi(0)=0$ such that
$G(x)=\Phi(F(x))$ in a neighbourhood of $x=0$.

\item[(ii)] The origin is a centre for system (\ref{lienard}) if and
only if there is a function $z(x)$ satisfying $z(0)=0$, $z^{\prime}(0)<0$ such
that $F(z)=F(x)$ and $G(z)=G(x)$.
\end{itemize}
\end{lemma}

The third approach, and the one which is of particular interest to
us here,  is the possibility of finding an \textit{integrating
factor}. If the origin is a critical point of focus type then it
is a centre if there is a function $D\neq0$ such that
\begin{equation}
\frac{\partial}{\partial x}\left(  DP\right)
+\frac{\partial}{\partial y}\left(  DQ\right)  =0 \label{dulac}
\end{equation}
in a neighbourhood of the origin. Such a function is called an
integrating factor or \textit{Dulac function}. The existence of
the  function $D$ means the system is integrable and the origin is
a centre.

We make a systematic search for an integrating factor of the form
$D=\Pi_{i=1}^{n}C_{i}^{\alpha_{i}}$, where each $C_{i}$ is an
invariant algebraic function. In this context an invariant
function is such that $\dot{C_{i}}=C_{i}L_{i}$, where $L_{i}$,
known as the cofactor of $C_{i}$, is of degree one less than that
of the system.  We require
\begin{equation}
\frac{\partial}{\partial x}\left(  DP\right)
+\frac{\partial}{\partial y}\left(  DQ\right)
=D(P_{x}+Q_{y}+\alpha_{1}L_{1}+\dots
+\alpha_{n}L_{n})=0.\label{dulacn}
\end{equation}
The $\alpha_{i}$ and the coefficients in the $C_{i}, L_{i}$ are
functions of the  coefficients $k,a_{i}$. We note that the
$C_{i},L_{i},\alpha_{i}$ may be complex; a real Dulac function is
then constructed from these together with their conjugates.  In
any given situation there may well be no invariant functions and
even though there is an upper bound for the possible degree of an
invariant curve it is not known how to determine this bound.
Darboux \cite{darboux} showed that if $n\geq \frac{1}{2}m(m+1)+2$
invariant functions exist, where $m$ is the degree of the system,
then the $n$ functions can be combined to form a first integral.
In practice we find that fewer such functions are required.  As
will be apparent later, finding such functions is non-trivial.
However it is a relatively straightforward matter to confirm that
the functions found actually satisfy the relation \eqref{dulac}.

These techniques for finding centre conditions are well
established  but the computational problems encountered are often
formidable.  We are constantly pushing the available software to
its limits.  The reduction of the focal values to obtain the
Liapunov quantities is one area which causes difficulties and here
we demonstrate the usefulness of our suite of programs INVAR
\cite{invar} in the search for invariant functions.  We are unable
to complete the reduction the focal values for \eqref{kukplus} to
obtain the conditions that are necessary for the origin to be a
centre, but we can find sufficient conditions by searching for
invariant functions.  We then try to determine whether or not we
have a complete set of conditions for the origin to be a centre.

The necessary and sufficient conditions for the origin to a centre
for  the Kukles system are known; we summarise them in Theorem
\ref{kuklestheorem}.  We note that the condition given in
\cite[Theorem 3.3]{kukles1}, with $a_{2}=0$, is covered by
condition (v) of Theorem \ref{kuklestheorem}. In \cite{kukles2} it
was conjectured that, when $a_{7}\neq 0$, the origin is a centre
for the Kukles system if and only if one of the conditions given
in Theorems 2.1 or 2.2 therein is satisfied.  This was verified in
\cite{kuklespersist}.

\begin{theorem}\label{kuklestheorem}
Let $\lambda =k=0$.  The origin is a centre for system
\eqref{kukplus}  if and only if one of the following conditions
holds:
\begin{itemize}
\item[(i)] $a_{2}=a_{5}=a_{7}=0$;

\item[(ii)] $a_{1}=a_{3}=a_{5}=a_{7}=0$;

\item[(iii)] $a_{4}=a_{3}(a_{1}+a_{3})$,
 $ a_{5}=-a_{2}(a_{1}+a_{3})$,
 $(a_{1}+2a_{3})a_{6}+a_{3}^2(a_{1}+a_{3})=0$, $a_{7}=0$;

\item[(iv)] $a_{5}+3a_{7}+a_{2}(a_{1}+a_{3})=0$,
 $9a_{6}a_{2}^2+2a_{2}^4+27a_{7}\mu +9\mu ^2=0$,
 $a_{4}a_{2}^2+a_{5}\mu =0$,
 $(3a_{7}\mu +\mu ^2+a_{6}a_{2}^2)a_{5}-3a_{7}\mu ^2-a_{6}a_{2}^2\mu=0$,
 where $\mu =3a_{7}+a_{2}a_{3}$;

\item[(v)] $a_{5}+3a_{7}+a_{2}(a_{1}+a_{3})=0$,
 $18a_{4}a_{5}-27a_{4}a_{7}+9a_{5}a_{1}^2+9a_{5}a_{6}+2a_{5}a_{2}^2=0$,
 $27a_{4}a_{1}+4a_{5}a_{2}+9a_{1}^3+2a_{1}a_{2}^2=0$,
 $18a_{4}^2+9a_{4}a_{1}^2+2a_{4}a_{2}^2+2a_{5}^2=0$,
 $18a_{4}a_{2}+9a_{5}a_{1}+9a_{5}a_{3}+9a_{1}^2a_{2}-27a_{1}a_{7}
  +9a_{6}a_{2}+2a_{2}^3=0$.

\end{itemize}
\end{theorem}

For a proof of the above theorem, see
\cite{cjcnglkukles,kukles1,kukles2,kuklespersist}.

This is one particular sub-class of system \eqref{kukplus}. In the
next section we shall present conditions that are necessary  and
sufficient for the origin to be a centre for two other sub-classes
of system \eqref{kukplus}; one with $a_{7}=0$ and one with
$a_{2}=0$. Presenting the results in this way allows for a clearer
description of the general case and gives us insight into the
types of invariant functions we should seek for system
\eqref{kukplus} in general.  In section $3$ we derive sufficient
conditions for the origin to be a centre for system
\eqref{kukplus} and in section $4$ we investigate whether there
are any other conditions. The coexistence of fine foci and the
bifurcation of small amplitude limit cycles is considered in
section $5$, and in section $6$ we investigate the possibility of
the existence of large amplitude limit cycles.

\section{Sub-classes $a_{7}=0$ and $a_{2}=0$}

In this section we consider the sub-classes of system
\eqref{kukplus} with $a_{7}=0$ or $a_{2}=0$.  We find that the
origin is a fine focus  of maximum order six when $a_{7}=0$ and
maximum order seven when $a_{2}=0$.

\begin{theorem}\label{a7=0}
Let $\lambda=a_{7}=0$. The origin is a centre for system
\eqref{kukplus} if and only if one of the following conditions holds:
\begin{itemize}
\item[(i)] $a_{2}=a_{5}=a_{7}=0$;

\item[(ii)] $k=a_{1}=a_{3}=a_{5}=a_{7}=0$;

\item[(iii)] $k=-\frac{1}{2}a_{1}$, $a_{3}=-\frac{2}{3}a_{1}$,
$a_{4}=-\frac {1}{4}a_{1}^{2}$, $a_{5}=-\frac{1}{3}a_{1}a_{2}$,
$a_{6}=a_{7} =0$;

\item[(iv)] $\ k=-a_{1}$, $a_{3}=-\frac{2}{3}a_{1}$,
$a_{4}=0$, $a_{5}=-\frac{1}{3}a_{1}a_{2}$, $a_{6}=a_{7}=0$;

\item[(v)] $\ k=-\frac{1}{4}a_{1}$, $a_{3}=-\frac{3}{4}a_{1}$,
$a_{4} =-\frac{1}{4}a_{1}^{2}$,  $a_{5}=-\frac{1}{4}a_{1}a_{2}$,
$a_{6} =a_{7}=0$;

\item[(vi)] $k=-\frac{1}{2}a_{1}$, $a_{3}=-a_{1}$,
$a_{5}=a_{6}=a_{7}=0$;

\item[(vii)] $a_{4}=a_{3}(a_{1}+a_{3})$, $a_{5}=-a_{2}(a_{1}+a_{3})$,
$(a_{1}+2a_{3})a_{6}-a_{3}(k-a_{3})(a_{1}+a_{3})=0$, $a_{7}=0$;

\item[(viii)] $k=-(a_{1}+a_{3})$, $a_{6}=a_{1}(a_{1}+a_{3})$,
$a_{5}=-a_{2}(a_{1}+a_{3})$,
$(3a_{1}+2a_{3})a_{4}+a_{1}^{2}(a_{1}+a_{3})=0$, $a_{7}=0$.

\end{itemize}
\end{theorem}

\begin{proof}
Calculation of the focal values for system \eqref{kukplus} with
$a_{7}=0$,  up to $\eta_{14}$, and their reduction to give the
corresponding Liapunov quantities is routine.  We do not present
the details here.  We find that $L(0)=L(1)=\dots=L(6)=0$ only if
one of the conditions of Theorem \ref{a7=0} holds.  The
sufficiency of these conditions is confirmed as follows.

When  (i) holds the system is invariant under the transformation
$(x,y,t)\mapsto(x,-y,-t)$; the system is symmetric in the $x$-axis, hence the
origin is a centre. Similarly, when condition (ii) holds the system is
invariant under the transformation $(x,y,t)\mapsto(-x,y,-t)$; the system is
symmetric in the $y$-axis, so the origin is a centre.

Conditions (iii), (iv), (v) and (vi) have $a_{6}=a_{7}=0$, in
which case  system \eqref{kukplus} is of the form
\begin{equation}
\dot{x}=(1+kx)y,\quad\dot{y}=x(-1+a_{1}x+a_{4}x^{2})+x(a_{2}+a_{5}
x)y+a_{3}y^{2}. \label{a6=0}
\end{equation}
If $k=0$ in these cases then condition (ii) is satisfied.  When $k\neq0$, we are able to transform \eqref{a6=0} to a Li\'{e}nard
system. The required transformation (see \cite{CLPCherkas}) is
$(x,y,t)\mapsto(x,(1+kx)y\Psi(x),\tau)$, where
\[
\Psi(x)=\frac{dt}{d\tau}=(1+kx)^{-1}\exp\Big(-\int_{0}^{x}a_{3}(1+ks)^{-1}
ds\Big)=(1+kx)^{-1-\frac{a_{3}}{k}}.
\]
 Then system \eqref{a6=0} becomes a system of the form \eqref{lienard} with
\[
f(x)=-x(a_{2}+a_{5}x)(1+kx)^{-1-\frac{a_{3}}{k}},\quad
g(x)=x(1-a_{1} x-a_{4}x^{2})(1+kx)^{-1-\frac{2a_{3}}{k}}.
\]
We compute the integrals of $f$, $g$ and denote these by $F$, $G$
respectively. For condition (iii) we have
\begin{gather*}
F(x)  =\frac{a_{2}}{a_{1}^{2}}\Big(  9-\big(  \frac{2}{a_{1}x-2}\big)
^{4/3}(a_{1}x-3)^{2}\Big)  ,\\
G(x)  =-\frac{6}{a_{1}^{2}}\Big(  3+\big(  \frac{2}{a_{1}x-2}\big)
^{2/3}\big(  a_{1}x-3\big)  \Big)  .
\end{gather*}
Let $u^{3}=a_{1}x-2$ and $v^{3}=a_{1}z-2$ then
\begin{gather*}
F(x)-F(z)   =\frac{2^{4/3}a_{2}}{a_{1}^{2}u^{4}v^{4}}(v-u)(u^{3}
v^{2}+u^{2}v^{3}-u^{2}-v^{2})\Omega,\\
G(x)-G(z)   =3\frac{2^{\frac{5}{3}}a_{2}}{a_{1}^{2}u^{2}v^{2}}(v-u)\Omega,
\end{gather*}
where
\[
\Omega=u^{2}v^{2}+u+v=\left(  \left(  a_{1}x-2\right)  \left(  a_{1}z-2\right)  \right)
^{2/3}+\left(  a_{1}x-2\right)  ^{1/3}+\left(  a_{1}
z-2\right)^{1/3}.
\]
When $x=z=0,\Omega_{x}=\Omega_{z}=-2^{-\frac{2}{3}}a_{1}$. By the
Implicit Function Theorem there is $z(x)$ with $z^{\prime}(x)<0$
such that $F(x)=F(z(x))$, $G(x)=G(z(x))$. The origin is a centre
by Lemma \ref{lemma1} (ii).
\par
Similarly for condition (iv) we find
\[
F(x)=\frac{a_{2}}{4a_{1}^{2}}\Big(  9-\frac{\left(  a_{1}x-3\right)^{2}
}{\left(  1-a_{1}x\right)  ^{2/3}}\Big)  ,\quad G(x)=-\frac
{3}{2a_{1}^{2}}\Big(  9+\frac{\left(  a_{1}x-3\right)}{\left(
1-a_{1}x\right)^{1/3}}\Big)
\]
and
$$\Omega=\left(\left(
1-a_{1}x\right)\left(1-a_{1}z\right)\right)^{1/3}\left(
\left(1-a_{1}x\right)^{1/3}+\left(1-a_{1}z\right)^{\frac{1}
{3}}\right)-2.$$\\
When condition (v) holds
\begin{gather*}
F(x)=-8\frac{a_{2}x^{2}}{(a_{1}x-4)^{2}},\\
G(x)=8\frac{x^{2}}
{(a_{1}x-4)^{6}}\left(  a_{1}^{4}x^{4}-24a_{1}^{3}x^{3}+240a_{1}^{2}
x^{2}-768a_{1}x+768\right)
\end{gather*}
and $\Omega=a_{1}xz-2(x+z)$.  For condition (vi)
\begin{gather*}
F(x)=-2\frac{a_{2}x^{2}}{(a_{1}x-2)^{2}},\\
G(x)=-4\frac{(a_{1}^2a_{4}x^4+4a_{1}^2x^2-8a_{1}x+4)}{(a_{1}x-2)^{4}}
\end{gather*}
and $\Omega=a_{1}xz-x-z$.  In each case $\Omega$ is a common factor
of $F(x)-F(z)$ and $G(x)-G(z)$; the origin is a centre by
Lemma \ref{lemma1} (ii).

To prove the sufficiency of the remaining conditions we use INVAR to help
us find appropriate invariant functions and to build Dulac functions.
Confirmation that the functions obtained are indeed Dulac functions
is routine.  When condition (vii) holds we find the Dulac function
\[
D=(1+kx)^{\alpha_{1}}e^{\alpha_{2}x}C^{\alpha_{3}},
\]
where
\begin{gather*}
C=1+a_{3}x-\gamma y,\quad
\alpha_{1}  =\frac{(a_{2}-2\gamma)(a_{3}k-a_{6})-k^{2}\gamma}{k^{2}\gamma},\\
\alpha_{2}=\frac{a_{6}(a_{2}-2\gamma)}{k\gamma},\quad
 \alpha_{3}=-\frac{a_{2}}{\gamma},
\end{gather*}
and $\gamma$ satisfies
$\gamma^{2}-a_{2}\gamma-a_{3}^2+a_{3}k-a_{6}=0$.   Hence, when
$k\gamma\neq0$, the origin is a centre.  When $k=0$ condition
(iii) of Theorem \ref{kuklestheorem} holds.  When $\gamma = 0$,
then $a_{6}=a_{3}(k-a_{3})=0$ and the system can be transformed to
Li\'{e}nard form with $f(x)=-a_{2}g(x)$; the origin is a centre by
Lemma \ref{lemma1} (i).

For condition (viii) we find the Dulac function
\[
D=(1+kx)^{\alpha_{1}}e^{\alpha_{2}x}C^{\alpha_{3}},
\]
where
\begin{gather*}
C =1-a_{1}x+\frac{a_{2}}{\gamma}y-a_{4}x^{2}+\frac{a_{5}}{\gamma}xy,\\
\alpha_{1}  =1, \quad  \alpha_{2}=a_{1}(\gamma+2), \quad \alpha_{3}=\gamma,
\end{gather*}
and $\gamma$ is a root of $a_{1}^{3}\gamma^{2}-(3a_{1}+2a_{3})a_{2}
^{2}(\gamma+1)=0$.  If $\gamma\neq0$, the origin is a centre.
When $\gamma =0$, then one of conditions (i), (ii) or (vii) is satisfied.
This completes the proof.
\end{proof}

When none of the conditions of Theorem \ref{a7=0} holds and
$L(i)=0$, for $i=0,1,2,\dots,5$, then $L(6) \neq 0$; the origin is
then a fine focus of maximum order six and at most six small
amplitude limit cycles can be bifurcated from the origin.


We now consider the sub-class of system \eqref{kukplus} with
$a_{2}=0$ and $a_{3}a_{7}\neq0$.  We exclude the possibility that
$a_{3}=0$ because,  when $a_{2}=a_{3}=0$, the origin is a centre
for system \eqref{kukplus} only if $a_{7}=0$.

\begin{theorem}\label{a2=0}
Let $\lambda=a_{2}=0$, with $a_{3}a_{7}\neq0$. The origin is a
centre  for system \eqref{kukplus} if and only if one of the
following conditions holds:
\begin{itemize}
\item[(i)] $a_{2}=0$, $k =-(2a_{1}+a_{3})$,
$(a_{1}+2a_{3})a_{4}+a_{1}^{2}(a_{1}+a_{3})=0$, $a_{5}=-3a_{7}$,
$(a_{1}+2a_{3})a_{6}-2a_{1}(a_{1}+a_{3})(2a_{1}+a_{3})=0$,
$2(a_{1}+2a_{3})^{2}a_{7}^{2}+a_{1}^{3}(a_{1}+a_{3})^{2}(3a_{1}+2a_{3})=0$;

\item[(ii)] $a_{2}=0$, $k =-(a_{1}+a_{3})$,
$2a_{3}a_{4}+a_{1}(a_{1}+a_{3})(a_{1}+3a_{3})=0$, $a_{5}=-3a_{7}$,
$2a_{3}a_{6}-a_{1}(a_{1}+a_{3})(3a_{1}+5a_{3})=0$,
$4a_{3}^{2}a_{7}^{2}+ a_{1}(a_{1}+a_{3})^{4}(a_{1}+2a_{3})=0$.
\end{itemize}
\end{theorem}

\begin{proof}
When $a_{2}=0$ and $a_{3}a_{7}\neq0$ we find that
$L(0)=L(1)=\dots=L(7)=0$ only if one of the conditions of Theorem
\ref{a2=0} holds.  The sufficiency of these conditions is
confirmed by constructing integrating factors from invariant
functions.  Again we use INVAR to find these functions.  When
condition (i) holds there exists a Dulac function
\begin{equation*}
D=(1+kx)^{\alpha _{1}}e^{\alpha _{2}x}C^{-3},
\end{equation*}
where
\begin{gather*}
C=1-a_{1}x+\frac{a_{1}^{2}(a_{1}+a_{3})}{(2a_{1}+a_{3})}
x^{2}+a_{7}xy,\\
 \alpha _{1}=\frac{(a_{1}+a_{3})^{2}}{(2a_{1}+a_{3})^{2}}
,  \alpha _{2}=-\frac{a_{1}(a_{1}+a_{3})}{(2a_{1}+a_{3})},
\end{gather*}
and hence the origin is a centre. We note that when $2a_{1}+a_{3}=0$,
then $k=0$ and condition (v) of Theorem \ref{kuklestheorem} is satisfied.

The Dulac function for condition (ii) is somewhat more complicated.
It consists of an invariant line, an invariant conic, an invariant
degree three curve and an invariant exponential.  We have
\begin{equation}
D=(1+kx)^{\alpha_{1}}e^{\alpha_{2}x}C_{1}^{\alpha_{3}}C_{2}^{\alpha_{4}},
\label{dulaca2=0}
\end{equation}
with
\begin{gather*}
C_{1}=1-a_{1}x+\frac{2a_{3}a_{7}}{k^{2}}y+\frac{k\tau(2a_{3}-k)}{2a_{3}
}x^{2}-\frac{2a_{1}a_{7}}{k}xy+\frac{k^{2}\tau}{2a_{3}}y^{2},\\
\begin{aligned}
C_{2} &=1+\frac{\Gamma_{1}}{12a_{3}^{2}\gamma^{2}w\upsilon}x-\gamma y
 +\frac{\Gamma_{2}}{72a_{3}^{5}\gamma^{2}w^{2}\upsilon
}x^{2}+\frac{\tau\Gamma_{3}}{8a_{3}^{3}\gamma^{2}w\upsilon}xy
+\frac{\Gamma_{4}}{12a_{3}^{2}\gamma^{2}\upsilon}y^{2}\\
&\quad +\frac{\tau(2a_{3}-k)\Gamma_{5}}{144a_{3}^{5}\gamma^{2}w^{2}\upsilon
}x^{3}+\frac{\Gamma_{6}}{48a_{3}^{5}\gamma^{2}w^{2}\upsilon}x^{2}y+\frac{\Gamma
_{7}}{36a_{3}^{3}\gamma^{2}w\upsilon}xy^{2}+wy^{3},
\end{aligned}
\\
\alpha_{1}  =\frac{9a_{3}k^{2}\rho\Phi_{0}+3k^{2}\rho\Phi_{1}\gamma-a_{3}
\Phi_{2}\gamma^{2}-6a_{3}k^{2}\Phi_{3}\gamma^{3}-4a_{3}^{2}k^{2}\rho\gamma^{4} (\Phi_{4}
-a_{3}a_{7} \gamma^{2})}{-48a_{3}^{4}\gamma^{2}w^{2}(3a_{3}a_{7}+k^{2}\gamma)},
\\
\alpha_{2}  =\frac{9a_{3}k^{3}\rho F_{0}+3k^{3}\rho F_{1}\gamma+a_{3}\tau
F_{2}\gamma^{2}-6a_{3}k^{2}\tau F_{3}\gamma^{3}-4a_{3}^{2}k^{3}\rho \gamma^{4} (F_{4}
- a_{3}a_{7} \gamma^{2})}{48a_{3}^{4}\gamma^{2}w^{2}(3a_{3}a_{7}+k^{2}\gamma)},
\\
\alpha_{3}  =-\frac{6a_{3}a_{7}\gamma}{3k^{2}\rho+4a_{3}a_{7}\gamma},  \alpha
_{4}=-\frac{3k^{2}\rho}{3k^{2}\rho+4a_{3}a_{7}\gamma},
\end{gather*}
where $\rho =a_{3}^{2}-k^{2}$, $\tau=a_{3}+k$ and
$\upsilon=4a_{3}^{2}a_{7}\gamma-3k^{3}\rho$.  Here $\gamma,w$ are roots of
\begin{gather*}
4a_{3}^{2}\gamma^{4}-36a_{1}a_{3}(a_{1}^{2}+a_{1}a_{3}-a_{3}
^{2})\gamma^{2}+81a_{1}^{2}k^{4}   =0,\\
\begin{aligned}
&64a_{3}^{6}w^{4}-16a_{1}^{3}a_{3}^{3}\big(a_{1}^{2}+a_{1}a_{3}-a_{3}^{2}\big)\\
&\times \big(a_{1}^{4}-4a_{1}^{3}a_{3}-22a_{1}^{2}a_{3}^{2}-20a_{1}a_{3}^{3}
+a_{3}^{4}\big)w^{2}+a_{1}^{6}k^{12}   =0,
\end{aligned}
\end{gather*}
respectively and the $\Gamma_{i},\Phi_{i}$ and $F_{i}$ are as given in
the Appendix.

To complete the proof we consider what happens when any of the
denominators  in the above are zero.  When $k \gamma w=0$, then
$a_{7}=0$. When $\upsilon = 0$ then either $a_{7}=0$ or
$a_{1}^2+7a_{1}a_{3}+8a_{3}^2=0$.  Let
$a_{1}=\frac{1}{2}(\sqrt{17}-7)a_{3}$.  We find a Dulac function
that consists of an invariant exponential function and three
invariant lines.  We have
\[
D=(1+kx)e^{\alpha_{1}x}C_{1}^{\alpha_{2}}C_{2}^{\alpha_{3}},
\]
with
\begin{gather*}
C_{1}=1+\dfrac{(4a_{3}^2+\vartheta^2)(5\vartheta^4
 +329a_{3}^2\vartheta ^2-4a_{3}^4)}{2\vartheta^2a_{3}\delta}x-\vartheta y,\\
C_{2}=1+\dfrac{\vartheta^4\Phi_{1}}{16a_{3}^3 \delta \beta}x-ny,\quad
\alpha_{1} =\dfrac{\vartheta^2 \Phi_{2}}{4a_{3}\Phi_{3}},\\
\alpha_{2}=-\dfrac{12a_{3}^2 \beta}{\varpi},\quad
 \alpha_{3}=\dfrac{12\vartheta a_{3}^2\beta}{n\varpi}
\end{gather*}
where $\delta=812a_{3}^2+33\vartheta^2$,
$\beta=4a_{3}^4+39a_{3}^2\vartheta^2 -73\vartheta^4$,
$\varpi=16a_{3}^6-301a_{3}^2\vartheta^4+5\vartheta^6$,
\[
n=\frac{4a_{3}^2\beta}{\vartheta(156a_{3}^4+9a_{3}^2\vartheta^2-5\vartheta^4)},
\]
$\vartheta$ is a root of
$(16a_{3}^4-32a_{3}^2\vartheta^2-\vartheta^4)(4a_{3}^4
-103a_{3}^2\vartheta^2-\vartheta^4)=0$ and $\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$ are polynomials of degree six in
$a_{3},\vartheta$.

When $(3a_{3}a_{7}+k^{2} \gamma)(3k^2\rho +4a_{3}a_{7}\gamma )=0$ then either $a_{7}=0$ or $2a_{1}+3a_{3}=0$.  Let $a_{1}=-\frac{3}{2}a_{3}$.  Then there exists a Dulac function
\[
D=(1+kx)C_{1}^{-2}C_{2}^{-1}
\]
where
\begin{gather*}
C_{1}=1+\frac{3}{4}a_{3}x+\frac{4a_{7}}{a_{3}}y,\\
C_{2}=1+\frac{3}{2}a_{3}x-\frac{8a_{7}}{a_{3}}y
+\frac{9}{16}a_{3}^2x^2-3a_{7}xy.
\end{gather*}
This completes the proof.
\end{proof}

When none of the conditions of Theorem \ref{a2=0} holds and
$L(i)=0$, for $i=0,1,2,\dots,6$, then $L(7) \neq 0$; the origin is
a fine focus of maximum order seven, at most seven small amplitude
limit cycles can be bifurcated from the origin.

\section{Sufficient centre conditions}

Now we return to the full system and derive some sufficient
conditions  for the origin to be a centre. We have obtained the
necessary and sufficient conditions for the origin to be a centre
for three sub-classes of system \eqref{kukplus}; with $k=0$, with
$a_{2}=0$ or with $a_{7}=0$. In these sub-classes we determined
possible centre conditions by considering the focal values and
then proved that the conditions we had found were sufficient.  As
the reduction of the focal values in the general case requires the
calculation of some resultants that cannot be obtained with the
currently available hardware and software we adopt a different
approach. We use the knowledge gained from consideration of the
sub-classes to give us an insight into the probable centre
conditions in the general case.  We search for invariant functions
and corresponding integrating factors for the general system
without introducing a condition for which the origin may be a
centre.  The relationships between the coefficients in system
\eqref{kukplus} that must be satisfied to ensure that
$\dot{C_{i}}=C_{i}L_{i}$ and  \eqref{dulacn} holds, for
$D=\Pi_{i=1}^{n}C_{i}^{\alpha_{i}}$, are sufficient conditions for
the origin to be a centre.  We find three sufficient conditions
for the origin to be a centre for system \eqref{kukplus}, with
$ka_{2}a_{7}\neq0$, using this approach.

Knowledge gained from the sub-classes suggests the type of
invariant  functions we should seek in order to determine
integrating factors. In particular, for the Kukles system and the
sub-class with $a_{7}=0$ combinations of invariant exponential
functions, invariant lines and invariant conics are required. The
Dulac functions for the class with $a_{2}=0$ are more complicated
and include invariant lines, conics and cubic functions. The line
$kx=-1$ is invariant with respect to system \eqref{kukplus}, with
$\lambda=0$, and is included in each Dulac function we seek in the
general case. Where the degrees of the equations in the system are
not equal it is often found that an exponential function is also
required and this is so in all cases here.

We search for functions that are invariant with respect to  system
\eqref{kukplus}.  Both $f(x)=e^{x}$ and $g(x)=kx+1$ are invariant
without any constraints on the coefficients $a_{i},k$.  Next we
look for functions that are invariant only when some relationships
between the coefficients are satisfied.  For the three sub-classes
we knew from the reduction of the focal values what these
relationships were.  Here we aim to find the relationships by
satisfying $\dot{C_{i}}=C_{i}L_{i}$ and equation \eqref{dulacn}.

We start with the simplest invariant curve, namely a line.   Let
$C=1+c_{10}x+c_{01}y$, with cofactor
$L=m_{10}x+m_{01}y+m_{20}x^2+m_{11}xy+m_{02}y^2$. We have
$c_{10}=m_{01}$ and $c_{01}=-m_{10}$; then seven equations must be
satisfied for $C=0$ to be invariant with respect to
\eqref{kukplus}.   We assume that $m_{10}\neq0$, otherwise we
recover the line $kx=-1$.  We determine
$m_{01},m_{20},m_{11},m_{02}$ in terms of $m_{10}$ and the
$a_{i},k$.  There are three remaining equations which must be
satisfied.  At this stage we try to build a Dulac function using
this line together with $f$ and $g$.   Five additional equations
must hold if $D=g^{\alpha_{1}}f^{\alpha_{2}}C^{\alpha_{3}}$ is a
Dulac function that satisfies \eqref{dulacn}. If $a_{7}\neq0$, we
must have $\alpha_{3}=-3$.  Then $m_{10}=\frac{1}{3}a_{2}$ and the
other $\alpha_{i}$ are given by two of these equations.  We have
determined all the coefficients of $C$ and $L$, and the
$\alpha_{i}$.  The four relationships between the coefficients
that must hold to satisfy the remaining equations are those of
condition (i) of Theorem \ref{sufficiency} below.

In a similar manner we search for invariant conics.   Let
$C=1+c_{10}x+c_{01}y+c_{20}x^2+c_{11}xy+c_{02}y^2$, with cofactor
$L$ as above.  Again $c_{10}=m_{01}$ and $c_{01}=-m_{10}$. Twelve
equations in the remaining eight coefficients of the conic and its
cofactor must be satisfied if $C=0$ is invariant with respect to
system \eqref{kukplus}.  Five additional equations must hold if
$D=g^{\alpha_{1}}f^{\alpha_{2}}C^{\alpha_{3}}$ is a Dulac function
that satisfies \eqref{dulacn}.  Consideration of all possible
situations in which the conic does not reduce to a line, or become
the product of two lines, leads us to conclude that either
$c_{02}=0$ or $m_{02}=2a_{7}$.  Let $c_{02}=0$.  If $a_{7}\neq 0$,
then $\alpha_{3}=-3$, the coefficients of the cofactor are
$m_{10}=\frac{1}{3}a_{2},m_{01}=-a_{1},m_{20}
=\frac{1}{3}a_{5},m_{11}=-a_{1}^2-a_{1}k-2a_{7}-\frac{2}{9}a_{2}^2$
and $C$, $\alpha_{1},\alpha_{2}$ are as given in the proof of
condition (ii) of Theorem \ref{sufficiency} below. As two of the
equations are linearly dependent in this situation this leaves
five equations in the coefficients $k,a_{i}$ that must be
satisfied if $\dot{C}=CL$ and \eqref{dulacn} holds.  These
equations lead to precisely the relationships of condition (ii) of
Theorem \ref{sufficiency}. When $c_{02}\neq 0$ and $m_{02}=2a_{7}$
these same five equations must be satisfied together with an
additional equation; this is a specific instance of condition
(ii).

We know that, when $a_{2}=0$, there is a Dulac function which
consists  of powers of $f,g$, an invariant conic and an invariant
cubic curve.  We search for this type of Dulac function in the
general case.  Again the linear coefficients in each invariant
curve can be given in terms of the linear coefficients in the
corresponding cofactor.  We have thirty-five equations in the
twenty-four unknowns.  In this instance the invariant conic in
which $c_{02}\neq 0$ and $m_{02}=2a_{7}$ is used.  We determine
all the coefficients of the invariant conic and its cofactor, in
terms of the coefficients of system \eqref{kukplus}, from the
twelve equations that must be satisfied for the conic to be
invariant with respect to \eqref{kukplus}.  This leaves four
relationships between the coefficients in \eqref{kukplus} that
must hold.

  We then proceed to determine the coefficients of the cubic function
and its cofactor.  Here there are eighteen equations in twelve
unknowns. We eliminate all but two of the unknowns, namely the
coefficient of $x$ in the cofactor (say $\gamma$) and the
coefficient of $y^3$ in the invariant cubic (say $w$).  One of the
remaining equations is quadratic in $\gamma$  and independent of
$w$.  Attempts to eliminate $\gamma$  from all remaining equations
using this equation lead to expressions being generated that
result in stack overflow.  We turn our attention to the five
equations that must be satisfied if \eqref{dulacn} holds.  We find
that if $a_{7}\neq 0$, then $\alpha_{3}=-\frac{3}{2}
(\alpha_{4}+1)$ and with $\alpha_{4}$ in terms of $\gamma,w$ and
the coefficients $k,a_{i}$ we must have
\begin{equation*}
(a_{1}a_{2}+a_{2}a_{3}+a_{5}+4a_{7})(a_{1}a_{2}+a_{2}a_{3}
+a_{5}+3a_{7})(a_{1}a_{3}+a_{3}^2-a_{4})=0.
\end{equation*}
The two remaining equations that must hold to satisfy
\eqref{dulacn} give $\alpha_{1},\alpha_{2}$.  We calculate that
$\eta_{4}=a_{1}a_{2}+a_{2}a_{3}+a_{5}+3a_{7}$; $\eta_{4}=0$ is
necessary for the origin to be a centre.  This, with the four
relationships from  the requirement for the conic to be invariant,
yield condition (iii) of Theorem \ref{sufficiency} below.  We use
these relationships to replace $k,a_{4},a_{5},a_{6},a_{7}$ in the
remaining equations.  We note that we have introduced another
unknown, $r$, where $r^2=a_{2}^2+4a_{3}^2-4k^2$.

We use the quadratic in $\gamma$  mentioned above to eliminate $r$ and,
 for consistency, we equate this expression for $r$ with
$\sqrt{a_{2}^2+4a_{3}^2-4k^2}$.  This consistency condition has
\begin{align*}
V=& (a_{2}^2 + 4a_{3}^2)\gamma ^4-3a_{2}(a_{2}^2 + 4a_{3}^2)\gamma ^3\\
& -9(4a_{1}^3a_{3} - 2a_{1}^2a_{2}^2 + 4a_{1}^2a_{3}^2 - 4a_{1}a_{2}^2a_{3}
  - 4a_{1}a_{3}^3 - a_{2}^2a_{3}^2)\gamma ^2\\
& -54a_{1}a_{2}(a_{1} + a_{3})^3\gamma +81a_{1}^2(a_{1} + a_{3})^4
\end{align*}
as a factor.  We know from consideration of a specific example that this
factor will ultimately lead to an appropriate Dulac function.
 This is the only remaining equation that is independent of $w$.

We factorise each of the equations and remove any factors that
involve only the remaining coefficients $a_{1},a_{2},a_{3}$; we
are able to show that such factors being zero lead to specific
instances of conditions that are already known to us.  Other than
$V=0$, the simplest of the remaining equations has over $7000$
terms.  We use a polynomial  remainder sequence to eliminate
$\gamma$  (see Section 4 for more details on polynomial remainder
sequences).  The later stages can only be completed by further
simplifying the expressions by replacing $a_{1}$ by $-(k+a_{3})$,
$a_{2}^2+4a_{3}^2$ by $t$ and scaling such that $k=1$.  For
example, at the second stage of the polynomial remainder sequence,
where a quadratic in $\gamma$  is produced with approximately
$30000$ terms, the size of the expressions can be almost halved by
these changes of variable.  We note however that in order to check
for factors that can be removed we need to replace $t$ by
$a_{2}^2+4a_{3}^2$ before attempting the factorisation.  The
calculations are repetitive, but formidable.  In some cases, in
order to multiply two expressions together, we have to split each
expression into smaller units and multiply each unit then sum the
results.  Near the final stage we produce an expression with
$191690$ terms, which we need to factorise.  Fortunately we can
predict that one of the factors will be the coefficient of
$\gamma^2$ at the quadratic stage of the polynomial remainder
sequence, an expression with $9411$ terms.  There are four other
factors, one of which is
\begin{align*}
W=& (a_{2}^2 + 4a_{3}^2)^3w^4+
 a_{2}(a_{2}^2 + 4a_{3}^2)^2(a_{2}^4 + 7a_{2}^2a_{3}^2
 - 6a_{2}^2k^2 + 12a_{3}^4 - 24a_{3}^2k^2 \\
&- 6a_{3}k^3 + 6k^4)w^3+
 (- a_{2}^6a_{3}^6 + 6a_{2}^6a_{3}^4k^2 + 6a_{2}^6a_{3}^3k^3
 - 3a_{2}^6a_{3}^2k^4  - 6a_{2}^6a_{3}k^5 \\
&- a_{2}^6k^6 - 12a_{2}^4a_{3}^8 +  72a_{2}^4a_{3}^6k^2 + 60a_{2}^4
  a_{3}^5k^3 - 69a_{2}^4a_{3}^4k^4 - 90a_{2}^4a_{3}^3k^5 \\
&+ 9a_{2}^4a_{3}^2k^6 + 36a_{2}^4a_{3}k^7 +
 6a_{2}^4k^8 - 48a_{2}^2a_{3}^{10} + 288a_{2}^2a_{3}^8k^2
 + 192a_{2}^2a_{3}^7k^3 \\
&-408a_{2}^2a_{3}^6k^4 - 432a_{2}^2a_{3}^5k^5 + 120a_{2}^2a_{3}^4k^6
 + 276a_{2}^2a_{3}^3k^7 + 72a_{2}^2a_{3}^2k^8 \\&- 12a_{2}^2a_{3}k^9
 - 64a_{3}^{12} +
 384a_{3}^{10}k^2 + 192a_{3}^9k^3 - 720a_{3}^8k^4 -  672a_{3}^7k^5 \\
&+ 272a_{3}^6k^6 + 528a_{3}^5k^7 + 192a_{3}^4k^8 + 16a_{3}^3k^9)w^2\\
& =a_{2}k^7(a_{3}+k)^3(6a_{2}^2a_{3}^2 + 3a_{2}^2a_{3}k - a_{2}^2k^2
 + 24a_{3}^4 + 12a_{3}^3k - 36a_{3}^2k^2 \\
&- 24a_{3}k^3)w +k^{12}(a_{3} + k)^6.
\end{align*}
We can show that when $V=W=0$ all remaining equations are satisfied.
We have found an appropriate Dulac function and condition (iii) of
Theorem \ref{sufficiency} is sufficient for the origin to be a centre.

\begin{theorem}\label{sufficiency}
Let $\lambda=0$.  The origin is a centre for system
\eqref{kukplus} if one of the following conditions holds:
\begin{itemize}
\item[(i)] $a_{5}=-a_{2}(a_{1}+a_{3})-3a_{7}$,\\[2mm]
$a_{6}=\dfrac{-2a_{2}^{4}-9a_{2}^{2}a_{3}^{2}+9a_{2}^{2}a_{3}k-81a_{2}
a_{3}a_{7}+27a_{2}a_{7}k-162a_{7}^{2}}{9a_{2}^{2}}$,\\[2mm]
$a_{4}=\dfrac{(-2a_{2}^{4}-9a_{2}^{2}a_{3}^{2}+9a_{2}^{2}a_{3}k-54a_{2}
a_{3}a_{7}+27a_{2}a_{7}k-81a_{7}^{2})\gamma^{2}}{2a_{2}^{6}}$,\\[2mm]
$a_{1}=\dfrac{(-4a_{2}^{4}-9a_{2}^{2}a_{3}^{2}+9a_{2}^{2}a_{3}k-54a_{2}
a_{3}a_{7}+27a_{2}a_{7}k-81a_{7}^{2})\gamma}{2a_{2}^{5}},\\[2mm]
\textrm{where }\gamma=a_{2}a_{3}+3a_{7}$ and $a_{2}\neq 0$;

\item[(ii)] $a_{5}=a_{2}k-3a_{7},\ k=-(a_{1}+a_{3}),$\\[2mm]
$a_{7}=\dfrac{k^2(a_{2}k-a_{3}r)}{a_{2}^2+4a_{3}^2}$,\\[2mm]
$a_{6}=\dfrac{k(a_{2}^2(a_{1}+3a_{3})-4a_{1}a_{3}(3a_{1}+5a_{3}))-3a_{2}k^2r}{2(a_{2}^2+4a_{3}^2)}$,\\[2mm]
$a_{4}=\dfrac{k(a_{2}^2(a_{1}-a_{3})+4a_{1}a_{3}(a_{1}+3a_{3}))+a_{2}k^2r}{2(a_{2}^2+4a_{3}^2)}$,\\[2mm]
where $r^2=a_{2}^2+4a_{3}^2-4k^2$ and $a_{2}^2+4a_{3}^2\neq 0$;

\item[(iii)] $a_{5}=-a_{2}(a_{1}+a_{3})-3a_{7}$,\\[2mm]
$a_{7}= \dfrac{9a_{1}^{2}(a_{1}+k)-2a_{2}^{2}(a_{1}+2a_{3})+9a_{4}(3a_{1}+2k)}
{12a_{2}}$,\\[2mm]
$a_{6}=\dfrac{9(18a_{4}k\epsilon-2a_{1}a_{2}^2\mu+9a_{1}(a_{1}^{2}(\mu+2k)
+a_{4}(3\mu+4k)+a_{1}k\epsilon))-8a_{2}^{2}\delta}{36a_{2}^{2}}$,\\[2mm]
$9(16a_{2}^{2}+(9a_{1}+6k)^{2})a_{4}^{2}+2 \rho (27a_{1}^{2}+18a_{1}k
 +4a_{2}^{2})a_{4}+a_{1}^{2} \rho ^{2}=0$,
\begin{align*}
&9(3a_{1}+2k)(4a_{2}^{2}+9 \mu (3a_{1}+2k))a_{4}^{2}
  +2\Big(81a_{1}^{2} \mu (a_{1}+k)(3a_{1}+2k)\\
&+36a_{1}a_{2}^{2}(2a_{1}+k)(a_{1}+k)-4a_{2}^{4}(a_{1}+2a_{3})\Big)a_{4}\\
&+a_{1}^{2} \rho \big(2a_{2}^{2}(k-a_{3})+9a_{1} \mu (a_{1}+k)\big)=0,
\end{align*}
where  $\rho =9a_{1}^{2}+9a_{1}k+2a_{2}^{2}$, $\mu =2a_{1}+a_{3}+k$,
$\epsilon=a_{3}+k$, $\delta=a_{2}^{2}+9a_{4}$ and $a_{2}\neq 0$.
\end{itemize}
\end{theorem}

\begin{proof}
When either condition (i) or (ii) holds we find a Dulac function
which consists of the line $kx=-1$, an exponential function and
either another line or a conic. The Dulac function then takes the
form $D=(1+kx)^{\alpha_{1}}e^{\alpha_{2}x}C^{-3}$, where $C$ and
the $\alpha_{i}$ are given below for each condition.  For
condition (iii) an invariant line, an invariant conic and an
invariant cubic together with an exponential function are needed.
\par
When condition (i) holds, and $k\neq0$, we find
\begin{gather*}
C =1+\frac{(a_{2}a_{3}+3a_{7})}{a_{2}}x-\frac{a_{2}}{3}y,\\
 \alpha_{1} =(2a_{2}^{4}+54a_{2}a_{7}k-81a_{7}^{2}+9(a_{3}^2-k^2)a_{2}^{2})/9a_{2}^{2}k^{2},\\
 \alpha_{2} =(-2a_{2}^{4}+27a_{2}a_{7}k+81a_{7}^{2}+9(k-a_{3})a_{2}^{2}
a_{3})/9a_{2}^{2}k.
\end{gather*}
When $k=0$, condition (iv) of Theorem \ref{kuklestheorem} holds.

For condition (ii) we find
\begin{gather*}
C =1-a_{1}x-\frac{a_{2}}{3}y-a_{4}x^{2}-\frac{a_{5}}{3}xy,\\
\alpha_{1} =(9a_{1}^{2}+2a_{2}^{2}-6a_{3}k+18a_{4}+6a_{6}-3k^{2})/3k^{2},\\
\alpha_{2} =-(9a_{1}^{2}+9a_{1}k+2a_{2}^{2}+18a_{4}+6a_{6})/3k,
\end{gather*}
with $k\neq0$. When $k=0$, condition (v) of Theorem
\ref{kuklestheorem} holds.

The Dulac function required when condition (iii) holds is by some way the
most complicated we have encountered. Here, in addition to the invariant
exponential function and the line $kx=-1$, we require an invariant conic,
$C_{1}=0$, and an invariant cubic, $C_{2}=0$. The Dulac function is
\[
D=(1+kx)^{\alpha_{1}}e^{\alpha_{2}x}C_{1}^{\alpha_{3}}C_{2}^{\alpha_{4}}.
\]
The invariant curves are not of high degree but have thousands of terms,
and the powers $\alpha_{i}$ are non-trivial.  The expressions are too
lengthy to be given here.  We note that when $a_{2}=0$, condition (iii)
becomes condition (ii) of Theorem \ref{a2=0} and the Dulac function
reduces to that of equation \eqref{dulaca2=0}.
\end{proof}

\section{Focal values}

Having established a set of sufficient conditions for  the origin
to be a centre for system \eqref{kukplus} with $ka_{2}a_{7} \neq
0$ we endeavour to ascertain if we have found the necessary
conditions.  If we could find a basis for the focal values for
system \eqref{kukplus} we would be able to determine the necessary
and sufficient conditions for the origin to be a centre. However
the computations soon become too large for the currently available
software and hardware systems.  We reduce the focal values as far
as is possible and, by using examples, determine whether or not
the sufficient conditions we have found are indeed the only
conditions for the origin to be a centre.  We conjecture that the
conditions given in Theorem \ref{sufficiency} are both necessary
and sufficient for the origin to be a centre for system
\eqref{kukplus}.

We calculate the focal values up to $\eta_{16}$ and in order to
simplify them we set $a_{1}=m-a_{3}$. We assume throughout this
section that $ka_{2}a_{7} \neq 0$.  We aim to establish under what
conditions the $L(i)$ are zero simultaneously. We have
\[
L(1)=a_{2}m+a_{5}+3a_{7}.
\]
Let $a_{5}=-a_{2}m-3a_{7}$. Then $L(1)=0$ and
$$L(2) =Aa_{6}+B,$$
where
$$A=a_{2}(a_{3}+m)-3a_{7}$$
and
\begin{align*}
B =&-2a_{2}^{2}a_{7}+2a_{2}a_{3}^{2}m-a_{2}a_{3}a_{4}-3a_{2}a_{3}km-5a_{2}a_{3}m^{2}+2a_{2}a_{4}k\\
&+5a_{2}a_{4}m+6a_{3}a_{7}m-9a_{4}a_{7}-9a_{7}km-15a_{7}m^{2}.
\end{align*}
Assume that $A\neq0$ and let $a_{6}=-B/A$. Then
\begin{gather*}
L(3)   =M_{0}+M_{1}a_{4}+M_{2}a_{4}^{2},\\
L(4)   =N_{0}+N_{1}a_{4}+N_{2}a_{4}^{2}+N_{3}a_{4}^{3},\\
L(5)   =P_{0}+P_{1}a_{4}+P_{2}a_{4}^{2}+P_{3}a_{4}^{3}+P_{4}a_{4}^{4},\\
L(6) =Q_{0}+Q_{1}a_{4}+Q_{2}a_{4}^{2}+Q_{3}a_{4}^{3}+Q_{4}a_{4}^{4}
+Q_{5}a_{4}^{5},
\end{gather*}
where the $M_{i}, N_{i}, P_{i}, Q_{i}$ are polynomials in
$k, a_{1} , a_{2}, a_{3}, a_{7}$.

In this instance calculating resultants to eliminate $a_{4}$ is
not feasible because of the degrees to which the variables occur
and the number of terms in the polynomials involved. We employ a
polynomial remainder sequence approach, the main advantage being
that we can work with the individual coefficients of the variable
being eliminated rather than the entire polynomial.  Also factors
of the reduced polynomials can be removed at each stage in the
process and some such factors can be predicted. We use the
following result to establish what these factors are.

\begin{lemma}
Suppose we have two univariate polynomials
$\alpha_{1}, \alpha_{2}$.  We can determine a sequence of
polynomials $\alpha_{3},\dots,\alpha_{j}$, of decreasing degree,
such that
\[
\mathop{\rm remainder}(\epsilon_{i}^{\delta_{i-1}+1}\alpha_{i-1},\alpha_{i})=\beta
_{i+1}\alpha_{i+1};\quad i\geqslant2,
\]
where $\delta_{i}$ is the difference in degrees between
$\alpha_{i}$ and $\alpha_{i+1};\,\epsilon_{i}$ is the leading
coefficient of
$\alpha_{i};\,\beta_{3}=1, \beta_{i+1}=\epsilon_{i-1}^{\delta_{i-2}+1}$.
Hence we have that $\beta_{i+1}$ divides the pseudo remainder of
$\alpha_{i-1}$ and $\alpha_{i}$.
\end{lemma}

The Proof of the above  lemma can be found in \cite{collins}.

Assume that $M_{2}\neq0$.  Let $a_{4}^{2}
=-(M_{0}+M_{1}a_{4})/M_{2}$ such that $L(3)=0$. Then
\begin{gather*}
L(4)   =A^{2}(\rho_{0}+\rho_{1}a_{4}),\\
L(5)   =A^{3}(\nu_{0}+\nu_{1}a_{4}),\\
L(6)   =A^{4}(\tau_{0}+\tau_{1}a_{4}),
\end{gather*}
where the $\rho_{i}, \nu_{i}, \tau_{i}$ are polynomials in
$m, a_{2}, a_{3}, a_{7}, k$. In particular
$\rho_{0}, \rho_{1}$\ are polynomials with $936,  654$ terms
respectively.

Assume that $\rho_{1}\neq0$ and let $a_{4}=-\rho_{0}/\rho_{1}$.
Then
\begin{gather*}
L(3)   =M_{2}^{2}A^{2}a_{7}\digamma\Omega,\\
L(5)   =M_{2}^{2}Aa_{7}\digamma\Gamma,\\
L(6)   =M_{2}^{2}Aa_{7}\digamma\Phi,
\end{gather*}
where
\begin{align*}
\digamma  =&-81a_{2}a_{7}^{2}k-54a_{2}^{2}a_{3}a_{7}k-9a_{2}^{3}a_{3}
^{2}k+243a_{7}^{3}+12a_{2}^{4}a_{7}\\
& +2a_{2}^{5}a_{1}+243a_{2}a_{3}a_{7}^{2}+4a_{2}^{5}a_{3}+81a_{2}^{2}a_{3}
^{2}a_{7}+9a_{2}^{3}a_{3}^{3}
\end{align*}
and $\Omega, \Gamma, \Phi$ are polynomials in
$m, a_{2}, a_{3}, a_{7}$.

When $\digamma=0$, the focal values $\eta_{8},\dots,\eta_{14}$
have a common factor
\begin{align*}
\Psi =& 2a_{2}^{6}a_{3}^{2}+2a_{2}^{6}a_{4}+12a_{2}^{5}a_{3}a_{7}+9a_{2}
^{4}a_{3}^{4}-9a_{2}^{4}a_{3}^{3}k+18a_{2}^{4}a_{7}^{2}+108a_{2}^{3}a_{3}
^{3}a_{7}\\
& -81a_{2}^{3}a_{3}^{2}a_{7}k+486a_{2}^{2}a_{3}^{2}a_{7}^{2}-243a_{2}^{2}
a_{3}a_{7}^{2}k+972a_{2}a_{3}a_{7}^{3}-243a_{2}a_{7}^{3}k+729a_{7}^{4}.
\end{align*}
Let
\begin{gather*}
a_{1}= \dfrac{(-4a_{2}^{4}-9a_{2}^{2}a_{3}^{2}+9a_{2}^{2}a_{3}k-54a_{2}
a_{3}a_{7}+27a_{2}a_{7}k-81a_{7}^{2})\gamma}{2a_{2}^{5}},\\
a_{4}=\dfrac{(-2a_{2}^{4}-9a_{2}^{2}a_{3}^{2}+9a_{2}^{2}a_{3}k-54a_{2}
a_{3}a_{7}+27a_{2}a_{7}k-81a_{7}^{2})\gamma^{2}}{2a_{2}^{6}},
\end{gather*}
where $\gamma=a_{2}a_{3}+3a_{7}$.  Then $\digamma=\Psi=0$ and
\[
a_{6}=\dfrac{-2a_{2}^{4}-9a_{2}^{2}a_{3}^{2}+9a_{2}^{2}a_{3}k-81a_{2}
a_{3}a_{7}+27a_{2}a_{7}k-162a_{7}^{2}}{9a_{2}^{2}}.
\]
These, together with $a_{5}=-a_{2}m-3a_{7}$, are condition (i) of
Theorem \ref{sufficiency}; the origin is a centre for system
\eqref{kukplus}. We note that in the special case when $A=B=0$
this condition is still satisfied and there are no other
conditions with $ka_{2}a_{7}\neq 0$.

The polynomials $\Omega, \Gamma, \Phi$ have $2294, 2895$ and $7674$ terms
respectively. The degrees to which each of the remaining variables occur in
$\Omega, \Gamma, \Phi$ are as shown in the following table:
\[
\begin{tabular}
[c]{|c|ccccc|} \hline
& $a_{2}$ & $a_{3}$ & $m$ & $a_{7}$ & $k$  \\ \hline
$\Omega$ & $12$ & $13$ & $19$ & $11$ & $11$\\
$\Gamma$ & $13$ & $14$ & $20$ & $12$ & $12$\\
$\Phi$ & $18$ & $19$ & $25$ & $15$ & $16$ \\
\hline
\end{tabular}
\]
Clearly any further progress in the reduction of the focal values
is going to be difficult, if not impossible, but we note that
$\Omega=\Gamma =\Phi=0 $ if either of the conditions (ii) or (iii)
of  Theorem \ref{sufficiency} holds.
\par
Suppose that we could calculate the resultants of
$\Omega, \Gamma$  and $\Omega, \Phi$ with respect to $a_{3}$.
Any common factor of the leading coefficients of $a_{3}$ in
$\Omega,\ \Gamma,\ \Phi$ will be a factor of the resultants, but
this common factor being zero may not be sufficient for the
vanishing of the polynomials.  We have that $a_{2}a_{7}m(k+m)$ is
such a factor.  In particular, when $k=-m$, we find
\[
\Upsilon_{1} = (a_{2}^2+4a_{3}^2)a_{7}^2+2a_{2}m^3a_{7}+a_{1}m^4(a_{1}+2a_{3})
\]
is a common factor of $\Omega,\ \Gamma$ and $\Phi$. Let
 \[
\omega=a_{7}^2=\frac{-2a_{2}m^3a_{7}-a_{1}m^4(a_{1}+2a_{3})}{a_{2}^2
+4a_{3}^2}.
\]
Then $L(3)=q_{0}+q_{1}a_{7}$.  Assume $q_{1}\neq 0$ and let
$a_{7}=-q_{0}/q_{1}$.  For consistency we must have
$\Upsilon=q_{1}^{2}\omega-q_{0}^2=0$.  We find
\[
\Upsilon_{2}= (a_{2}^2+4a_{3}^2)a_{4}^2+(4a_{1}a_{3}(a_{1}+3a_{3})+a_{2}^2(a_{1}-a_{3}))ma_{4}
+a_{1}m^2(a_{1}(a_{1}+3a_{3})^2-a_{2}^2a_{3})
\]
is a common factor of $L(4)$ and $\Upsilon$.  Now $k=-m$,
 $a_{5}=-a_{2}m-3a_{7}$, $a_{6}=-B/A$,
 $\Upsilon_{1}= \Upsilon_{2}=0$ is precisely condition (iii)
of Theorem \ref{sufficiency}; the origin is a centre.

We have seen how conditions (i) and (iii) of Theorem \ref{sufficiency}
emerge from the reduction of the focal values.  However we are unable
to locate condition (ii) by a similar argument.  It is possible that
conditions (ii) and (iii) are specific instances of more general conditions.
 We show that this is not the case by considering a particular example.
Each of these conditions has five relationships between the eight
coefficients $a_{i},k$.  We can choose values for three of the
variables  without imposing new relationships.

Let $a_{1}=1$, $a_{2}=1$, $a_{3}=-2$.  Now $\Omega,\Gamma,\Phi$
are  polynomials in $a_{7},k$, and $a_{4}$, $a_{5}$, $a_{6}$ are
given in terms of $a_{7},k$ also.  Let $R(f,g,x)$ denote the
resultant of $f$ and $g$ with respect to $x$ and \# represent a
(large) integer. We calculate resultants with respect to $a_{7}$
and find
\begin{gather*}
R(\Omega,\Gamma,a_{7})=\#(k-1)^3(2k-3)^6(k^2+6k+10)^4\phi K_{1}K_{2},\\
R(\Omega,\Phi,a_{7})=\#(k-1)^3(2k-3)^4(k^2+6k+10)^6\phi
K_{1}K_{3},
\end{gather*}
where $\phi=162k^3+152k^2-28k-84$ and $K_{1},K_{2},K_{3}$ are irreducible
polynomials of degrees $52,64,95$ in $k$, respectively.  When $K_{1}=0$,
then $\rho_{0}=\rho_{1}=0$ and when $2k-3=0$, then $a_{7}=0$; both situations
are excluded under current assumptions.
The leading coefficients of $a_{7}$ in $\Omega,\Gamma,\Phi$ have $k^2+6k+10$,
which is positive definite, as a common factor.  For the general case this
factor is $a_{2}^2+(a_{1}-a_{3}+k)^2$, which is non-zero when $a_{2}\neq 0$.
 Clearly $K_{2},K_{3}$ cannot be zero simultaneously.  So we must have
$k=1(=-m)$ or $\phi=0$; the former is true when condition (iii) holds,
the latter when condition (ii) holds.  Furthermore there are no other centre
conditions with five or fewer relationships that satisfy
$\Omega=\Gamma=\Phi=0$.

After extensive consideration of all the possible situations in which
the Liapunov quantities up to $L(6)$ are zero simultaneously we have not
found any centre conditions other than those given in
Theorems \ref{kuklestheorem}, \ref{a7=0}, \ref{a2=0} and \ref{sufficiency}.
 However there are several instances where we are unable to calculate
resultants or eliminate variables using a polynomial remainder
sequence, so we cannot be certain that we have a complete set of
necessary conditions.   We have shown, by considering an example,
that when $\Omega=\Gamma=\Phi=0$ there are no other such
conditions which contain five or fewer relationships between the
coefficients.  We can also demonstrate that when one of $k,a_{2}$
or $a_{7}$ is zero in $\Omega,\Gamma,\Phi$ then all the centre
conditions found, by considering $\Omega=\Gamma=\Phi=0$, can be
obtained from one of the conditions in Theorem \ref{sufficiency}
with the appropriate variable set to zero.  This wealth of
evidence leads us to the following claim.

\begin{conjecture} \label{conj7}
The origin is a centre for system \eqref{kukplus} when $ka_{2}a_{7}\neq 0$
if and only if one of the conditions of Theorem \ref{sufficiency} holds.
\end{conjecture}

\section{Coexisting fine foci}

One of the significant features of a planar dynamical system is the possible
configuration of limit cycles. Information is sought on the number of
critical points that can be encircled by closed orbits and on the number of
closed orbits that can encircle one such critical point. This may be phrased
as asking how many nests of limit cycles can there be and how many limit
cycles make up each nest.
\par
Suppose that $\lambda=0$ in system \eqref{kukplus}, so that the
origin is a fine focus. From the equation for $\dot{x}$, critical
points can occur only on the $x$-axis and on $kx=-1$. However
$kx=-1$ is invariant, so any critical point on it cannot be of
focus type. Thus any fine focus must have $y=0$ and
$x(a_{4}x^{2}+a_{1}x-1)=0$. The condition for a fine point is
$x(a_{5} x+a_{2})=0$. Thus only one critical point other than the
origin can be a fine focus.

\begin{lemma} \label{lem8}
System \eqref{kukplus} with $\lambda=0$ can have at most two fine foci.
\end{lemma}

Suppose that there are two fine foci. We can scale coordinates so that they
are $(0,0)$ and $(1,0)$. Then $a_{4}=1-a_{1},a_{5}=-a_{2}$ and the system is
\begin{equation} \label{coexist2}
\begin{gathered}
\dot{x}=y(1+kx),\\
\dot{y}=-x+a_{1}x^{2}+a_{2}xy+a_{3}y^{2}+(1-a_{1}
)x^{3}-a_{2}x^{2}y+a_{6}xy^{2}+a_{7}y^{3}.
\end{gathered}
\end{equation}
The point $(1,0)$ is a fine focus, as opposed to a fine col, if
$(k+1)(a_{1}-2)>0$.  We denote the Liapunov quantities associated
with the origin by $L(i)$ and those for the point $(1,0)$ by
$M(i)$.

\begin{theorem}\label{sixone}
Suppose that the origin and $(1,0)$ coexist as fine foci in
system \eqref{coexist2}. If $(1,0)$ is of order one then the origin is of
order at most six.
\end{theorem}

\begin{proof}
We calculate that
\begin{gather}
L(1)=a_{2}(a_{1}+a_{3}-1)+3a_{7}, \label{L(1)} \\
M(1)=3(a_{1}-2)^{2}a_{7}+a_{2}((2-a_{1})a_{6}+(a_{1}-1)(k-a_{3}+1)+a_{3}
).\label{M(1)}
\end{gather}
For $(1,0)$ to be a fine focus of order one we must ensure that
$M(1)\neq0$. Let $a_{7}=\frac{1}{3}a_{2}(1-a_{1}-a_{3})$, then
$L(1)=0$ and
\[
L(2)=a_{2}(3a_{6}(2a_{1}+3a_{3}-1)+\varphi),
\]
where
\begin{align*}
\varphi =&15a_{1}^{3}+24a_{1}^{2}a_{3}+9a_{1}^{2}k-39a_{1}^{2}+2a_{1}
a_{2}^{2}+9a_{1}a_{3}^{2}+9a_{1}a_{3}k-45a_{1}a_{3}\\
& -15a_{1}k+33a_{1}+2a_{2}^{2}a_{3}-2a_{2}^{2}-9a_{3}^{2}-9a_{3}
k+21a_{3}+6k-9.
\end{align*}
If $a_{2}=0$, then $a_{5}=a_{7}=0$ and the origin is a centre by Theorem
\ref{a7=0} (i). We require $a_{2}\neq0$ for the origin to be a fine focus
of order greater
than two. If $2a_{1}+3a_{3}=1$ then $\varphi
=a_{2}\Phi$, where
\begin{equation}
\Phi=(a_{1}-2)(2a_{2}^{2}+9(a_{1}-1)^{2})+9(k+1)(a_{1}
-1)^{2},\label{phi}
\end{equation}
which is non-zero if $a_{2}\neq0$ and $(1,0)$ is a fine focus.

So, for the origin to be a fine focus of order greater than two,
we also need $2a_{1}+3a_{3}\neq1$.  Let
$a_{6}=\frac{\varphi}{3(1-2a_{1}-3a_{3})}$. Now we require
$M(1)=a_{2}(a_{1}+a_{3}-1)\Phi$ to be non-zero and we have
\[
L(3)=M(1)(\Gamma-2a_{2}^{2}(a_{1}+a_{3}-1)),
\]
where
\begin{align*}
\Gamma =&3(25a_{1}^{3}+97a_{1}^{2}a_{3}+7a_{1}^{2}k-32a_{1}^{2}+117a_{1}
a_{3}^{2}+22a_{1}a_{3}k-88a_{1}a_{3}\\
& -4a_{1}k+14a_{1}+45a_{3}^{3}+15a_{3}^{2}k-54a_{3}^{2}-8a_{3}k+22a_{3}-2).
\end{align*}
Let $a_{2}^{2}=\frac{\Gamma}{2(a_{1}+a_{3}-1)}$. Then
\[
L(4)=a_{2}(A k^{2}+B k+C)\Omega,
\]
where
\begin{gather*}
A  = 15(a_{1}+a_{3}) ^{2}-20(a_{1}+a_{3}) +8>0,\\
B  = 15(a_{1}+a_{3}) ^2(11a_{1}+13a_{3})-5(a_{1}+a_{3}) (59a_{1}+67a_{3})
+4(45a_{1}+49a_{3}-8),\\
\begin{aligned}
C = &90(a_{1}+a_{3}) ^3(4a_{1}+5a_{3})-45(a_{1}+a_{3}) ^2(19a_{1}+23a_{3})\\
&+30(a_{1}+a_{3}) (25a_{1}+29a_{3})-8(35a_{1}+38a_{3}-5),
\end{aligned}\\
\begin{aligned}
\Omega= &15(a_{1}-2)a_{3}^2+(a_{1}-2)(29a_{1}+5k-13)a_{3}+14a_{1}^{3}
 +5a_{1}^{2}k-40a_{1}^{2}\\
&-11a_{1}k+28a_{1}+3k-7.
\end{aligned}
\end{gather*}
When $a_{3}$ is either root of $\Omega=0$, we have $a_{2}^{2}<0$.
We require $a_{2}$ to be real, so $\Omega\neq0$. As $A$ is
non-zero let $k=\frac{(-B+r)}{2A }$, where
\begin{equation}
r=\sqrt{B ^{2}-4AC}.\label{disc}
\end{equation}
Then
\[
L(5)=a_{2}\Omega(\alpha r+\beta),
\]
where $\alpha,\beta$ are polynomials of degrees nine and twelve
in $a_{1}, a_{3}$. Assume for the time being that $\alpha\neq0$.
Let $r=-\frac{\beta}{\alpha}$. Then
\[
L(6)=a_{2}\Omega(3a_{1}+3a_{3}-2)^{2}\Theta,
\]
where $\Theta$ is a polynomial of degree sixteen in $a_{1}$ and
$a_{3}$. For consistency, $r$ as given by \eqref{disc} must also
be equal to $-\dfrac{\beta}{\alpha}$. This is true if
$(3a_{1}+3a_{3}-2)\Lambda=0$, where $\Lambda$ is a polynomial of
degree twelve in $a_{1}, a_{3}$. When $3a_{1}+3a_{3}=2$ then
$k=-1$, and $(1,0)$ is not a fine focus. The origin can be a fine
focus of order greater than six only if $\Theta=\Lambda=0$. We
calculate the resultants of $\Theta$ and $\Lambda$ with respect to
$a_{1}$ and $a_{3}$. We find
\begin{align*}
R(\Theta,\Lambda,a_{3})
=& \#(a_{1}-2)^{3}(a_{1}-1)^{3}(a_{1}-5)^{2}(2a_{1}-1)(6a_{1}^{3}-58a_{1}^{2}+183a_{1}-191)\\
& \times(8a_{1}^{2}
-41a_{1}+53)(27a_{1}^{4}-180a_{1}
^{3}+478a_{1}^{2}-620a_{1}+343)^{2}\digamma\Upsilon,
\end{align*}
where $\digamma, \Upsilon$ are polynomials of degrees eighteen
and forty in $a_{1}$. The quadratic and degree four factors are
positive definite. The one real root of the cubic factor, together
with the corresponding value for $a_{3}$, are such that
$a_{1}+a_{3}=1$, which is excluded if $(1,0)$ is a fine focus of
order one. When $a_{1}=\frac{1}{2}$ (or $a_{1}=5$) then $a_{3}=0$
(or $a_{3}=-3$) and $2a_{1}+3a_{3}=1$, which is also excluded.
Similarly, when $a_{1}=1$ or $a_{1}=2$ then $(k+1)(a_{1} -2)\leq
0$.

Using Sturm sequences \cite{sturm} we find that $\Upsilon=0$ has
six distinct real roots. We also locate the corresponding roots
$a_{3}$, and find that all the root pairs are such that
$(k+1)(a_{1}-2)\leq 0$. When $\digamma=0$, we have
$\alpha=\beta=0$. A similar analysis to that for the case
$\alpha\neq0$ leads us to conclude that, when $\alpha=0$, the
origin can be of maximum order five when $(1,0)$ is of order one.
\par
We conclude that $L(6)\neq0$ under current assumptions; the origin is of
maximum order six and at most seven small amplitude limit cycles can be
 bifurcated simultaneously from the two fine foci.
\end{proof}

\begin{theorem} \label{thm10}
Suppose that the origin and $(1,0)$ coexist as fine foci for system
\eqref{coexist2}. If $(1,0)$ is of order greater than one then both the fine
foci are of maximum order two, or both are centres.
\end{theorem}

\begin{proof}
We have $L(1),M(1)$ given by \eqref{L(1)}, \eqref{M(1)}
respectively. Again we make a substitution for $a_{7}$ from
$L(1)=0$. Then
\[
M(1)=a_{2}(a_{6}(2-a_{1})+\phi),
\]
where
\[
\phi=(1-a_{1})(a_{1}^{2}+a_{1}a_{3}-4a_{1}-2a_{3}-k+3).
\]
If $(1,0)$
is a fine focus then $a_{1}\neq2$.  Let $a_{6}=\frac{\phi}{a_{1}-2}$.  Then
\[
L(2)=M(2)=a_{2}(a_{1}-2)(a_{1}+a_{3}-1)\Phi,
\]
where $\Phi$ is given by \eqref{phi}.  If $a_{2}=0$, then
$a_{5}=a_{7}=0$ and both critical points are centres by Theorem
\ref{a7=0} (i). Similarly, when $a_{1}+a_{3}=1$ then $a_{4}=a_{3},
a_{6}=\frac{a_{3}(k-a_{3})}{a_{3}+1}, a_{7}=0$ and both critical
points are centres by Theorem \ref{a7=0} (iv). If $a_{2}\neq 0$
and $(1,0)$ is a fine focus then $(a_{1}-2)\Phi\neq 0$.  If $L(2)$
is non-zero, so is $M(2)$ . Hence both points are fine foci of
maximum order two or they are both centres.
\end{proof}

We can demonstrate that four small amplitude limit cycles can be
bifurcated from the two fine foci of order two of system
\eqref{kukplus}.  We begin with system \eqref{kukplus} with
\[
\lambda=0, \quad a_{4}=1-a_{1},\quad  a_{5}=-a_{2}, \quad
a_{7}=-\frac{1}{3}a_{2}(a_{1}+a_{3}-1), \quad
a_{6}=\frac{\phi}{a_{1}-2}.
\]
Hence the origin and $(1,0)$ are fine foci of order two, and
$L(0)=M(0)=L(1)=M(1)=0,M(2)=L(2)\neq 0$.  First we perturb $a_{6}$
such that $M(1)$ becomes non-zero and of opposite sign to $M(2)$.
If $(2-a_{1})(a_{1}+a_{3}-1)>0$ we decrease $a_{6}$, otherwise we
increase $a_{6}$.  The stability of $(1,0)$ is reversed and a
limit cycle bifurcates.  Next we perturb $a_{7}$ such that $L(1)$
becomes non-zero and of opposite sign to $L(2)$, so reversing the
stability of the origin.  If $a_{2}(a_{1}+a_{3}-1)>0$ then we
decrease $a_{7}$, otherwise we increase $a_{7}$.  A second limit
cycle bifurcates, but this time from the origin.  Another limit
cycle can be bifurcated from the origin, by increasing $\lambda$
if $a_{2}(a_{1}+a_{3}-1)>0$ or decreasing $\lambda$ otherwise.  To
bifurcate a fourth limit cycle from $(1,0)$ we require the
stability of $(1,0)$ to be reversed, so then it has the same
stability as the origin.  Hence we require $\lambda M(1)<0$, which
is the case when $M(1)M(2)<0$.

Similarly seven small amplitude limit cycles can be bifurcated
from  the two fine foci when one is of order one and the other is
of order six.

\section{Large amplitude limit cycles}

We have proved that the origin can be a fine focus of order seven
and we  have investigated the possibility of small amplitude limit
cycles bifurcating from two coexisting fine foci.  In this case we
found that the maximum number of small amplitude limit cycles that
can exist simultaneously is seven.  By considering the global
phase portrait within a particular parameter range, we shall
demonstrate that a large amplitude limit cycle can surround two
fine foci in system \eqref{kukplus}.  It is known that the Kukles
system, with two fine foci of order two, can have a large
amplitude limit cycle surrounding both critical points
\cite{wuchenyang}.

\begin{theorem} \label{thm11}
If the fine foci at the origin and $(1,0)$ are both of order two
for system \eqref{kukplus} then at least five limit cycles exist
under  certain conditions.
\end{theorem}

\begin{proof}
We begin with two fine foci, each of maximal order two.
Therefore system \eqref{kukplus} is of the form
\begin{equation} \label{22dist}
\begin{gathered}
\dot{x}=y\left( 1+kx\right), \\
\dot{y}=-x+a_{1}x^{2}+a_{2}xy+(1-a_{1}+\delta
)y^{2}+(1-a_{1})x^{3}-a_{2}x^{2}y+Axy^{2}-\dfrac{a_{2}\delta }{3}
y^{3},
\end{gathered}
\end{equation}
where
$$
\delta =a_{1}+a_{3}-1, \quad A=(1-a_{1})\Big(\delta -\dfrac{(a_{1}+k-1)}{
(a_{1}-2)}\Big),\quad  (k+1)(a_{1}-2)>0.
$$
One can consider system \eqref{22dist} as a perturbation of the system
\begin{equation} \label{a2perturb}
\begin{gathered}
\dot{x}=y\left( 1+kx\right), \\
\dot{y}=-x+a_{1}x^{2}+(1-a_{1}+\delta
)y^{2}+(1-a_{1})x^{3}+Axy^{2},
\end{gathered}
\end{equation}
with the introduction of the term
\begin{equation*}
\beta=-a_{2}y\Big( (x-\frac{1}{2})^{2}+\frac{\delta }{3}y^{2}-\frac{1}{4}\Big),
\end{equation*}
when $a_{2}$ is perturbed from zero.  System \eqref{a2perturb} has
centres at the origin and $(1,0)$, and a col at
$(\frac{1}{a_{1}-1},0)$. We consider a particular global phase
portrait of system \eqref{a2perturb}.  We can arrange for there to
be no critical points on the line $kx=-1$ by choosing values of
the parameters such that
\begin{equation*}
k(a_{1}-2)(\delta -a_{1})-k+a_{1}^2(\delta-1)+a_{1}(2-3\delta)+2\delta -1\geq0.
\end{equation*}
For example, we can take $k=1,a_{1}=3,\delta=3$.  We then use
polar  coordinates to consider the critical points at infinity.
Provided that $A\leq 0$, that is $\delta \geq
\frac{a_{1}+k-1}{a_{1}-2}$, the only critical points at infinity
lie on $\theta=\pm \frac{\pi}{2}$.

At infinity $\dot{\theta}\leq0$, so the motion is clockwise in the
region  $kx>-1$ and the outward separatrices of the col cannot
tend to a critical point at infinity.  The system is symmetric in
the $x$-axis, so the separatrices form homoclinic loops and the
orbits outside the `figure of eight' so formed are closed.  Take
one of these closed orbits; $\Gamma$, say.  Increase $a_{2}$ so
that system \eqref{a2perturb} becomes system \eqref{22dist}.  For
system \eqref{22dist} the flow is inwards across $\Gamma$ because
the vector product of the two fields is
\begin{equation*}
-y^2(1+kx)a_{2}\Big( (x-\frac{1}{2})^{2}+\frac{\delta }{3}y^{2}-\frac{1}{4}\Big).
\end{equation*}
If $a_{2}>0$, the two fine foci are both unstable and hence there is a large
amplitude limit cycle inside $\Gamma$.

In the previous section we demonstrated that a total of four small
amplitude  limit cycles can be bifurcated from the origin and
$(1,0)$ simultaneously.  Our current assumptions are consistent
with the argument therein.  Therefore system \eqref{kukplus} can
have five limit cycles.
\end{proof}

This leads us to the following conjecture.

\begin{conjecture} \label{conj12}
When system \eqref{kukplus} has two fine foci of orders six and one
than at least eight limit cycles can exist.
\end{conjecture}

\subsection*{Concluding remarks}
We have presented various
properties of system \eqref{kukplus}.  In particular we have found
sufficient conditions for the origin to be a centre by finding
complicated invariant functions that can be combined to form a
Dulac function. We conjecture that we have found the necessary and
sufficient conditions for the origin to be a centre for system
\eqref{kukplus} even though we were unable to complete the
reduction of the focal values because of the size of the
expressions generated.  We also proved some results on the
possible configurations of limit cycles.

\section{Appendix}

The polynomials required in the proof of condition (ii) of Theorem
\ref{a2=0} are as follows:
\begin{align*}
\Phi_{0} =& a_{3}^{4}a_{7}k^{4}+2a_{3}^{3}a_{7}k^{5}+2a_{3}^{3}k^{4}
w+4a_{3}^{2}a_{7}w^{2}+2a_{3}^{2}k^{5}w\\
&-2a_{3}a_{7}k^{7}-2a_{3}k^{6}w-a_{7}k^{8}-2k^{7}w,
\\
\Phi_{1} =&2a_{3}^{6}k^{4}+20a_{3}^{5}a_{7}w+2a_{3}^{5}k^{5}-5a_{3}^{4}
k^{6}+12a_{3}^{4}w^{2}-24a_{3}^{3}a_{7}k^{2}w-6a_{3}^{3}k^{7}
\\
&-12a_{3}^{3}kw^{2}-4a_{3}^{2}a_{7}k^{3}w+2a_{3}^{2}k^{8}+4a_{3}k^{9}+k^{10},
\\
\Phi_{2} =&-4a_{3}^{8}a_{7}k^{2}-12a_{3}^{7}k^{2}w+16a_{3}^{6}a_{7}
k^{4}+12a_{3}^{6}k^{3}w+4a_{3}^{5}a_{7}k^{5}+30a_{3}^{5}k^{4}w\\
&-21a_{3}^{4}a_{7}k^{6}-72a_{3}^{4}a_{7}w^{2}-24a_{3}^{4}k^{5}w-10a_{3}^{3}a_{7}k^{7}+216a_{3}
^{3}a_{7}kw^{2}\\
&-24a_{3}^{3}k^{6}w+8a_{3}^{2}a_{7}k^{8} +12a_{3}^{2}k^{7}w+6a_{3}a_{7}k^{9}+6a_{3}k^{8}w+a_{7}k^{10},\\
\Phi_{3}=&a_{3}^{5}k^{4}-8a_{3}^{4}a_{7}w+a_{3}^{4}k^{5}-2a_{3}^{3}
k^{6}+8a_{3}^{3}w^{2}+8a_{3}^{2}a_{7}k^{2}w\\
&-2a_{3}^{2}k^{7}+a_{3}k^{8}+k^{9},\\
\Phi_{4}=&2a_{3}^{3}a_{7}+3a_{3}^{2}w-3a_{3}a_{7}k^{2}-3a_{3}kw-a_{7}k^{3},
\end{align*}
\begin{align*}
F_{0}
=&a_{3}^{4}a_{7}k^{4}+2a_{3}^{3}a_{7}k^{5}+2a_{3}^{3}k^{4}w+4a_{3}
^{2}a_{7}w^{2}+2a_{3}^{2}k^{5}w-2a_{3}a_{7}k^{7}-2a_{3}k^{6}w\\
&-a_{7}k^{8}-2k^{7}w,\\
F_{1}
=&2a_{3}^{6}k^{4}+20a_{3}^{5}a_{7}w+2a_{3}^{5}k^{5}+12a_{3}^{4}
a_{7}kw-5a_{3}^{4}k^{6}+12a_{3}^{4}w^{2}-12a_{3}^{3}a_{7}k^{2}w\\
& -6a_{3}^{3}k^{7}-4a_{3}^{2}a_{7}k^{3}w+2a_{3}^{2}k^{8}+4a_{3}k^{9}+k^{10},\\
F_{2} =&4a_{3}^{7}a_{7}k^{3}-4a_{3}^{6}a_{7}k^{4}+12a_{3}^{6}k^{3}
w-12a_{3}^{5}a_{7}k^{5}-12a_{3}^{5}k^{4}w+8a_{3}^{4}a_{7}k^{6}\\
&+144a_{3}^{4}a_{7}w^{2}-18a_{3}^{4}k^{5}w+13a_{3}^{3}a_{7}k^{7}-216a_{3}^{3}a_{7}kw^{2}+12a_{3}
^{3}k^{6}w\\
&-3a_{3}^{2}a_{7}k^{8}+6a_{3}^{2}k_{7}w-5a_{3}a_{7}k^{9}-a_{7}k^{10},\\
F_{3}=&a_{3}^{4}k^{5}-8a_{3}^{3}a_{7}kw+4a_{3}^{3}w^{2}+8a_{3}^{2}
a_{7}k^{2}w-2a_{3}^{2}k^{7}+k^{9},\\
F_{4} =&2a_{3}^{3}a_{7}+3a_{3}^{2}w-3a_{3}a_{7}k^{2}-a_{7}k^{3},
\end{align*}
\begin{align*}
\Gamma_{1} =&96a_{3}^{6}a_{7}k^{3}\gamma
^{2}-32a_{3}^{6}k^{4}\gamma ^{3}+12a_{3}
^{5}a_{7}k^{4}\gamma ^{2}+144a_{3}^{5}a_{7}\gamma ^{3}w-9a_{3}^{5}k^{7}\gamma \\
& -108a_{3}^{5}k^{3}\gamma ^{2}w-243a_{3}^{4}a_{7}k^{7}-282a_{3}^{4}a_{7}k^{5}
\gamma ^{2}-36a_{3}^{4}a_{7}k^{3}\gamma w+72a_{3}^{4}k^{8}\gamma \\
& +98a_{3}^{4}k^{6}\gamma ^{3}+12a_{3}^{4}k^{4}\gamma ^{2}w-486a_{3}^{3}a_{7}
k^{8}-114a_{3}^{3}a_{7}k^{6}\gamma ^{2}+180a_{3}^{3}k^{9}\gamma \\
& +34a_{3}^{3}k^{7}\gamma ^{3}+108a_{3}^{3}k^{5}\gamma ^{2}w+186a_{3}^{2}a_{7}k^{7}
\gamma ^{2}+36a_{3}^{2}a_{7}k^{5}\gamma w+18a_{3}^{2}k^{10}\gamma \\
& -66a_{3}^{2}k^{8}\gamma ^{3}-12a_{3}^{2}k^{6}\gamma ^{2}w+486a_{3}a_{7}k^{10}
+102a_{3}a_{7}k^{8}\gamma ^{2}-171a_{3}k^{11}\gamma \\
& -34a_{3}k^{9}\gamma ^{3}+243a_{7}k^{11}-90k^{12}\gamma ,
\end{align*}
\begin{align*}
\Gamma_{2} =&512a_{3}^{11}a_{7}k^{4}\gamma
^{3}-384a_{3}^{11}k^{7}\gamma ^{2}
-36a_{3}^{10}a_{7}k^{7}\gamma +2016a_{3}^{10}a_{7}k^{3}\gamma ^{2}w\\
&-96a_{3}^{10}k^{8}\gamma ^{2}-672a_{3}^{10}k^{4}\gamma ^{3}w-1152a_{3}^{9}a_{7}k^{8}\gamma -2624a_{3}^{9}a_{7}
k^{6}\gamma ^{3}\\
&+228a_{3}^{9}a_{7}k^{4}\gamma ^{2}w+768a_{3}^{9}a_{7}\gamma ^{3}w^{2}+837a_{3}^{9}k^{11}+2256a_{3}^{9}k^{9}
\gamma ^{2}\\
&-225a_{3}^{9}k^{7}\gamma w+96a_{3}^{9}k^{5}\gamma ^{3}w-576a_{3}^{9}k^{3}\gamma ^{2}w^{2}-2106a_{3}^{8}a_{7}k^{9}\gamma \\
&-1052a_{3}^{8}a_{7}k^{7}\gamma ^{3}-5211a_{3}^{8}a_{7}k^{7}w-5958a_{3}^{8}a_{7}k^{5}\gamma ^{2}w-540a_{3}^{8}a_{7}k^{3}\gamma w^{2}\\
&+1917a_{3}^{8}k^{12}+1206a_{3}^{8}k^{10}\gamma ^{2}+1575a_{3}^{8}k^{8}\gamma w+2058a_{3}^{8}k^{6}\gamma ^{3}w\\
&+180a_{3}^{8}k^{4}\gamma ^{2}w^{2}+2556a_{3}^{7}a_{7}k^{10}\gamma +4164a_{3}^{7}a_{7}k^{8}\gamma ^{3}-10071a_{3}^{7}a_{7}k^{8}w\\
&-2316a_{3}^{7}a_{7}k^{6}\gamma ^{2}w+1368a_{3}^{7}a_{7}k^{4}\gamma w^{2}+792a_{3}^{7}a_{7}k^{2}\gamma ^{3}w^{2}-288a_{3}^{7}a_{7}\gamma ^{2}w^{3}\\
&-1917a_{3}^{7}k^{13}-4572a_{3}^{7}k^{11}\gamma ^{2}+3726a_{3}^{7}k^{9}\gamma w+402a_{3}^{7}k^{7}\gamma ^{3}w\\
&-783a_{3}^{7}k^{7}w^{2}-126a_{3}^{7}k^{5}\gamma ^{2}w^{2}+216a_{3}^{7}k^{3}\gamma w^{3}+8226a_{3}^{6}a_{7}k^{11}\gamma \\
&+2936a_{3}^{6}a_{7}k^{9}\gamma ^{3}+702a_{3}^{6}a_{7}k^{9}w+4008a_{3}^{6}a_{7}k^{7}\gamma ^{2}w+2484a_{3}^{6}a_{7}k^{5}\gamma w^{2}\\
&-168a_{3}^{6}a_{7}k^{3}\gamma ^{3}w^{2}-7695a_{3}^{6}k^{14}-3462a_{3}^{6}k^{12}\gamma ^{2}-216a_{3}^{6}k^{10}\gamma w\\
&-1488a_{3}^{6}k^{8}\gamma ^{3}w-1458a_{3}^{6}k^{8}w^{2}-54a_{3}^{6}k^{6}\gamma ^{2}w^{2}+3528a_{3}^{5}a_{7}k^{12}\gamma \\
&-1672a_{3}^{5}a_{7}k^{10}\gamma ^{3}+10422a_{3}^{5}a_{7}k^{10}w+2088a_{3}^{5}a_{7}k^{8}\gamma ^{2}w+576a_{3}^{5}a_{7}k^{6}\gamma w^{2}\\
&-2187a_{3}^{5}k^{15}+3600a_{3}^{5}k^{13}\gamma ^{2}-3861a_{3}^{5}k^{11}\gamma w-480a_{3}^{5}k^{9}\gamma ^{3}w\\
&+108a_{3}^{5}k^{9}w^{2}+702a_{3}^{5}k^{7}\gamma ^{2}w^{2}-216a_{3}^{5}k^{5}\gamma w^{3}-4806a_{3}^{4}a_{7}k^{13}\gamma \\
&-1884a_{3}^{4}a_{7}k^{11}\gamma ^{3}+4509a_{3}^{4}a_{7}k^{11}w-66a_{3}^{4}a_{7}k^{9}\gamma ^{2}w+8667a_{3}^{4}k^{16}\\
&+3690a_{3}^{4}k^{14}\gamma ^{2}-1377a_{3}^{4}k^{12}\gamma w+102a_{3}^{4}k^{10}\gamma ^{3}w+1458a_{3}^{4}k^{10}w^{2}\\
&-126a_{3}^{4}k^{8}\gamma ^{2}w^{2}-4932a_{3}^{3}a_{7}k^{14}\gamma -380a_{3}^{3}a_{7}k^{12}\gamma ^{3}-351a_{3}^{3}a_{7}k^{12}w\\
&+6777a_{3}^{3}k^{17}-588a_{3}^{3}k^{15}\gamma ^{2} +360a_{3}^{3}k^{13}\gamma w-18a_{3}^{3}k^{11}\gamma ^{3}w+675a_{3}^{3}k^{11}
w^{2}\\
&-1278a_{3}^{2}a_{7}k^{15}\gamma -1917a_{3}^{2}k^{18} -1338a_{3}^{2}k^{16}\gamma ^{2}+18a_{3}^{2}k^{14}\gamma w-3510a_{3}k^{19}\\
&-312a_{3}k^{17}\gamma ^{2}-972k^{20},
\end{align*}
\begin{align*}
\Gamma_{3} =&192a_{3}^{8}k^{4}\gamma ^{2}-576a_{3}^{7}a_{7}\gamma
^{2}w-192a_{3}
^{7}k^{5}\gamma ^{2}-64a_{3}^{6}a_{7}k^{3}\gamma ^{3}+576a_{3}^{6}a_{7}k\gamma ^{2}w\\
& -432a_{3}^{6}k^{8}-780a_{3}^{6}k^{6}\gamma ^{2}+56a_{3}^{5}a_{7}k^{4}
\gamma ^{3}+1296a_{3}^{5}a_{7}k^{4}w+576a_{3}^{5}a_{7}k^{2}\gamma ^{2}w\\
& -432a_{3}^{5}k^{9}+390a_{3}^{5}k^{7}\gamma ^{2}+48a_{3}^{5}k^{3}\gamma ^{3}
w+162a_{3}^{4}a_{7}k^{7}\gamma +132a_{3}^{4}a_{7}k^{5}\gamma ^{3}\\
& +1296a_{3}^{4}a_{7}k^{5}w-48a_{3}^{4}a_{7}k^{3}\gamma ^{2}w+1323a_{3}^{4}
k^{10}+1134a_{3}^{4}k^{8}\gamma ^{2}\\
&-32a_{3}^{4}k^{4}\gamma ^{3}w +162a_{3}^{3}a_{7}k^{8}\gamma -56a_{3}^{3}a_{7}k^{6}\gamma ^{3}+48a_{3}^{3}a_{7}
k^{4}\gamma ^{2}w+1782a_{3}^{3}k^{11}\\
&-54a_{3}^{3}k^{9}\gamma ^{2} -16a_{3}^{3}k^{5}\gamma ^{3}w-162a_{3}^{2}a_{7}k^{9}\gamma -68a_{3}^{2}a_{7}k^{7}
\gamma ^{3}-432a_{3}^{2}k^{12}\\
&-546a_{3}^{2}k^{10}\gamma ^{2}-162a_{3}a_{7}k^{10}\gamma
 -1350a_{3}k^{13}-144a_{3}k^{11}\gamma ^{2}-459k^{14},
\end{align*}
\begin{align*}
\Gamma_{4} =&128a_{3}^{6}a_{7}\gamma ^{3}-96a_{3}^{6}k^{3}\gamma
^{2}+36a_{3}^{5}
a_{7}k^{3}\gamma -12a_{3}^{5}k^{4}\gamma ^{2}-288a_{3}^{4}a_{7}k^{4}\gamma \\
& -264a_{3}^{4}a_{7}k^{2}\gamma ^{3}-48a_{3}^{4}a_{7}\gamma ^{2}w+243a_{3}^{4}
k^{7}+282a_{3}^{4}k^{5}\gamma ^{2}+36a_{3}^{4}k^{3}\gamma w\\
& -684a_{3}^{3}a_{7}k^{5}\gamma -136a_{3}^{3}a_{7}k^{3}\gamma ^{3}+486a_{3}^{3}
k^{8}+114a_{3}^{3}k^{6}\gamma ^{2}-360a_{3}^{2}a_{7}k^{6}\gamma \\
& -186a_{3}^{2}k^{7}\gamma ^{2}-36a_{3}^{2}k^{5}\gamma w-486a_{3}k^{10}-102a_{3}k^{8}
\gamma ^{2}-243k^{11},
\end{align*}
\begin{align*}
\Gamma_{5} =&512a_{3}^{10}a_{7}k^{4}\gamma
^{3}-96a_{3}^{9}k^{8}\gamma ^{2}-480a_{3}
^{9}k^{4}\gamma ^{3}w-1116a_{3}^{8}a_{7}k^{8}\gamma \\
&-2624a_{3}^{8}a_{7}k^{6}\gamma ^{3}+1020a_{3}^{8}a_{7}k^{4}\gamma ^{2}w+192a_{3}^{8}a_{7}\gamma ^{3}w^{2}-96a_{3}^{8}
k^{9}\gamma ^{2}\\
&+96a_{3}^{8}k^{5}\gamma ^{3}w-2232a_{3}^{7}a_{7}k^{9}\gamma  -1088a_{3}^{7}a_{7}k^{7}\gamma ^{3}+144a_{3}^{7}a_{7}k^{5}\gamma ^{2}w\\
&+243a_{3}^{7}k^{12}+390a_{3}^{7}k^{10}\gamma ^{2}+1197a_{3}^{7}k^{8}\gamma w +1470a_{3}^{7}k^{6}\gamma ^{3}w\\
&-324a_{3}^{7}k^{4}\gamma ^{2}w^{2}+2412a_{3}^{6}a_{7}k^{10}\gamma +4128a_{3}^{6}a_{7}k^{8}\gamma ^{3} -1944a_{3}^{6}a_{7}k^{8}w\\
&-2748a_{3}^{6}a_{7}k^{6}\gamma ^{2}w+828a_{3}^{6}a_{7}k^{4}\gamma w^{2}+648a_{3}^{6}a_{7}k^{2}\gamma ^{3}w^{2}
-288a_{3}^{6}a_{7}\gamma ^{2}w^{3}\\
&+729a_{3}^{6}k^{13}+492a_{3}^{6}k^{11}\gamma ^{2}+2169a_{3}^{6}k^{9}\gamma w+216a_{3}^{6}k^{7}\gamma ^{3}w\\
&+108a_{3}^{6}k^{5}\gamma ^{2}w^{2}+8352a_{3}^{5}a_{7}k^{11}\gamma +3008a_{3}^{5}a_{7}k^{9}\gamma ^{3}-3888a_{3}^{5}a_{7}k^{9}w\\
&-996a_{3}^{5}a_{7}k^{7}\gamma ^{2}w+1944a_{3}^{5}a_{7}k^{5}\gamma w^{2}-24a_{3}^{5}
a_{7}k^{3}\gamma ^{3}w^{2}+243a_{3}^{5}k^{14}\\
&-390a_{3}^{5}k^{12}\gamma ^{2}-450a_{3}^{5}k^{10}\gamma w-1092a_{3}^{5}k^{8}\gamma ^{3}w+324a_{3}^{5}k^{6}\gamma ^{2}
w^{2}\\
&+3708a_{3}^{4}a^{7}k^{12}\gamma -1600a_{3}^{4}a_{7}k^{10}\gamma ^{3}+1728a_{3}^{4}a_{7}k^{8}\gamma ^{2}w+1116a_{3}
^{4}a_{7}k^{6}\gamma w^{2}\\
&-1215a_{3}^{4}k^{15} -696a_{3}^{4}k^{13}\gamma ^{2}-2394a_{3}^{4}k^{11}\gamma w-312a_{3}^{4}k^{9}
\gamma ^{3}w\\
&-108a_{3}^{4}k^{7}\gamma ^{2}w^{2}-4824a_{3}^{3}a_{7}k^{13}\gamma -1920a_{3}^{3}a_{7}k^{11}\gamma ^{3}+3888a_{3}^{3}a_{7}k^{11}w\\
&+852a_{3}^{3}a_{7}k^{9}\gamma ^{2}w-1215a_{3}^{3}k^{16}-6a_{3}^{3}k^{14}\gamma ^{2}-747a_{3}^{3}k^{12}\gamma w\\
&+102a_{3}^{3}k^{10}\gamma ^{3}w-5004a_{3}^{2}a_{7}k^{14}\gamma -416a_{3}^{2}a_{7}k^{12}\gamma ^{3}+1944a_{3}^{2}a_{7}k^{12}w\\
&+243a_{3}^{2}k^{17}+300a_{3}^{2}k^{15}\gamma ^{2}+225a_{3}^{2}k^{13}\gamma w-1296a_{3}a_{7}k^{15}\gamma \\
&+729a_{3}k^{18}+102a_{3}k^{16}\gamma ^{2}+243k^{19},
\end{align*}
\begin{align*}
\Gamma_{6} =&-3072a_{3}^{13}a_{7}k^{4}\gamma
^{2}+2880a_{3}^{12}k^{4}
\gamma ^{2}w+6912a_{3}^{11}a_{7}k^{8}+21888a_{3}^{11}a_{7}k^{6}\gamma ^{2}\\
&-1152a_{3}^{11}a_{7}\gamma ^{2}w^{2}+256a_{3}^{11}k^{7}\gamma ^{3}-576a_{3}^{11}k^{5}\gamma ^{2}w+13824a_{3}^{10}a_{7}
k^{9}\\
&+9408a_{3}^{10}a_{7}k^{7}\gamma ^{2}-960a_{3}^{10}a_{7}k^{3}\gamma ^{3}w+64a_{3}^{10}k^{8}\gamma ^{3}-6480a_{3}^{10}k^{8}w\\
&-14580a_{3}^{10}k^{6}\gamma ^{2}w-28512a_{3}^{9}a_{7}k^{10}-55704a_{3}^{9}a_{7}k^{8}\gamma ^{2} -488a_{3}^{9}a_{7}k^{4}\gamma ^{3}w\\
&+2592a_{3}^{9}a_{7}k^{4}w^{2}-1584a_{3}^{9}a_{7}k^{2}\gamma ^{2}w^{2}-558a_{3}^{9}k^{11}\gamma -1504a_{3}^{9}k^{9}\gamma ^{3}\\
& -11664a_{3}^{9}k^{9}w-2514a_{3}^{9}k^{7}\gamma ^{2}w+96a_{3}^{9}k^{3}\gamma ^{3}
w^{2}-85050a_{3}^{8}a_{7}k^{11}\\
&-44580a_{3}^{8}a_{7}k^{9}\gamma ^{2} +2394a_{3}^{8}a_{7}k^{7}\gamma w+2844a_{3}^{8}a_{7}k^{5}\gamma ^{3}w+5184a_{3}^{8}
a_{7}k^{5}w^{2}\\
&+648a_{3}^{8}a_{7}k^{3}\gamma ^{2}w^{2} -1278a_{3}^{8}k^{12}\gamma -804a_{3}^{8}k^{10}\gamma ^{3}+15957a_{3}^{8}k^{10}
w\\
&+24042a_{3}^{8}k^{8}\gamma ^{2}w+216a_{3}^{8}k^{4}\gamma ^{3}w^{2} -7614a_{3}^{7}a_{7}k^{12}+51984a_{3}^{7}a_{7}k^{10}\gamma ^{2}\\
&+5634a_{3}^{7}a_{7}k^{8}\gamma w+2264a_{3}^{7}a_{7}k^{6}\gamma ^{3}w+11340a_{3}^{7}a_{7}k^{6}w^{2}\\
&+7440a_{3}^{7}a_{7}k^{4}\gamma ^{2}w^{2}+192a_{3}^{7}a_{7}\gamma ^{3}w^{3}+1278a_{3}^{7}k^{13}\gamma  +3048a_{3}^{7}k^{11}\gamma ^{3}\\
&+42930a_{3}^{7}k^{11}w+9954a_{3}^{7}k^{9}\gamma ^{2}w+414a_{3}^{7}k^{7}\gamma w^{2}+156a_{3}^{7}k^{5}\gamma ^{3}w^{2}\\
& -144a_{3}^{7}k^{3}\gamma ^{2}w^{3}+137862a_{3}^{6}a_{7}k^{13}+66036a_{3}^{6}
a_{7}k^{11}\gamma ^{2}+1692a_{3}^{6}a_{7}k^{9}\gamma w\\
& -1520a_{3}^{6}a_{7}k^{7}\gamma ^{3}w+17172a_{3}^{6}a_{7}k^{7}w^{2}+2952a_{3}
^{6}a_{7}k^{5}\gamma ^{2}w^{2}+5130a_{3}^{6}k^{14}\gamma \\
& +2308a_{3}^{6}k^{12}\gamma ^{3}+8991a_{3}^{6}k^{12}w-12690a_{3}^{6}k^{10}
\gamma ^{2}w+972a_{3}^{6}k^{8}\gamma w^{2}\\
&-228a_{3}^{6}k^{6}\gamma ^{3}w^{2} +112914a_{3}^{5}a_{7}k^{14}-2520a_{3}^{5}a_{7}k^{12}\gamma ^{2}-4788a_{3}^{5}
a_{7}k^{10}\gamma w\\
&-1776a_{3}^{5}a_{7}k^{8}\gamma ^{3}w+8100a_{3}^{5}a_{7}k^{8}w^{2}-1104a_{3}^{5}a_{7}k^{6}\gamma ^{2}w^{2}
+1458a_{3}^{5}k^{15}\gamma \\
&-2400a_{3}^{5}k^{13}\gamma ^{3} -29808a_{3}^{5}k^{13}w-6462a_{3}^{5}k^{11}\gamma ^{2}w+144a_{3}^{5}k^{9}
\gamma w^{2}\\
&-252a_{3}^{5}k^{7}\gamma ^{3}w^{2}+144a_{3}^{5}k^{5}\gamma ^{2}w^{3} -30942a_{3}^{4}a_{7}k^{15}-29436a_{3}^{4}a_{7}k^{13}\gamma ^{2}\\
&-4086a_{3}^{4}a_{7}k^{11}\gamma w-364a_{3}^{4}a_{7}k^{9}\gamma ^{3}w-324a_{3}^{4}a_{7}k^{9}w^{2}-5778a_{3}^{4}k^{16}\gamma \\
&-2460a_{3}^{4}k^{14}\gamma ^{3}-17901a_{3}^{4}k^{14}
w+774a_{3}^{4}k^{12}\gamma ^{2}w-972a_{3}^{4}k^{10}\gamma w^{2}\\
& +12a_{3}^{4}k^{8}\gamma ^{3}w^{2}-78570a_{3}^{3}a_{7}k^{16}-12576a_{3}^{3}
a_{7}k^{14}\gamma ^{2}-846a_{3}^{3}a_{7}k^{12}\gamma w\\
&-4518a_{3}^{3}k^{17}\gamma  +392a_{3}^{3}k^{15}\gamma ^{3}-486a_{3}^{3}k^{15}w-402a_{3}^{3}k^{13}
\gamma ^{2}w\\
&-558a_{3}^{3}k^{11}\gamma w^{2}-35694a_{3}^{2}a_{7}k^{17} -1428a_{3}^{2}a_{7}k^{15}\gamma ^{2}+1278a_{3}^{2}k^{18}\gamma \\
&+892a_{3}^{2}k^{16}\gamma ^{3}-567a_{3}^{2}k^{16}w-426a_{3}^{2}k^{14}\gamma ^{2}w -5130a_{3}a_{7}k^{18}\\
&+2340a_{3}k^{19}\gamma +208a_{3}k^{17}\gamma ^{3}-972a_{3}k^{17}w+648k^{20}\gamma ,
\end{align*}
\begin{align*}
\Gamma_{7} =&768a_{3}^{9}a_{7}k^{3}\gamma
^{2}-256a_{3}^{9}k^{4}\gamma ^{3}+192a_{3}
^{8}a_{7}k^{4}\gamma ^{2}+384a_{3}^{8}a_{7}\gamma ^{3}w+18a_{3}^{8}k^{7}\gamma \\
& -288a_{3}^{8}k^{3}\gamma ^{2}w-1674a_{3}^{7}a_{7}k^{7}-3744a_{3}^{7}a_{7}
k^{5}\gamma ^{2}+180a_{3}^{7}a_{7}k^{3}\gamma w\\
& +576a_{3}^{7}k^{8}\gamma +1312a_{3}^{7}k^{6}\gamma ^{3}-492a_{3}^{7}k^{4}\gamma ^{2}
w-3834a_{3}^{6}a_{7}k^{8}\\
&-2220a_{3}^{6}a_{7}k^{6}\gamma ^{2} -936a_{3}^{6}a_{7}k^{4}\gamma w-792a_{3}^{6}a_{7}k^{2}\gamma ^{3}w-144a_{3}^{6}
a_{7}\gamma ^{2}w^{2}\\
&+1053a_{3}^{6}k^{9}\gamma +526a_{3}^{6}k^{7}\gamma ^{3} +783a_{3}^{6}k^{7}w+846a_{3}^{6}k^{5}\gamma ^{2}w+108a_{3}^{6}k^{3}\gamma w^{2}\\
&+2160a_{3}^{5}a_{7}k^{9}+5400a_{3}^{5}a_{7}k^{7}\gamma ^{2} -2304a_{3}^{5}a_{7}k^{5}\gamma w-264a_{3}^{5}a_{7}k^{3}\gamma ^{3}w\\
&+108a_{3}^{5}a_{7}k^{3}w^{2}-1278a_{3}^{5}k^{10}\gamma -2082a_{3}^{5}k^{8}\gamma ^{3}+2484a_{3}^{5}k^{8}w\\
&+1698a_{3}^{5}k^{6}\gamma ^{2}w-36a_{3}^{5}k^{4}
\gamma w^{2}+11556a_{3}^{4}a_{7}k^{10}+4704a_{3}^{4}a_{7}k^{8}\gamma ^{2}\\
& -1044a_{3}^{4}a_{7}k^{6}\gamma w-4113a_{3}^{4}k^{11}\gamma -1468a_{3}^{4}k^{9}
\gamma ^{3}+1836a_{3}^{4}k^{9}w\\
&-114a_{3}^{4}k^{7}\gamma ^{2}w -108a_{3}^{4}k^{5}\gamma w^{2}+6534a_{3}^{3}a_{7}k^{11}-1800a_{3}^{3}a_{7}
k^{9}\gamma ^{2}\\
&+180a_{3}^{3}a_{7}k^{7}\gamma w-144a_{3}^{3}a_{7}k^{5}\gamma ^{3}w -108a_{3}3a_{7}k^{5}w^{2}-1764a_{3}^{3}k^{12}\gamma \\
&+836a_{3}^{3}k^{10}\gamma ^{3}-1566a_{3}^{3}k^{10}w-1206a_{3}^{3}k^{8}\gamma ^{2}w+36a_{3}^{3}k^{6}\gamma w^{2}\\
&-5778a_{3}^{2}a_{7}k^{12}-2676a_{3}^{2}a_{7}
k^{10}\gamma ^{2}+36a_{3}^{2}a_{7}k^{8}\gamma w+2403a_{3}^{2}k^{13}\gamma \\
& +942a_{3}^{2}k^{11}\gamma ^{3}-2619a_{3}^{2}k^{11}w-444a_{3}^{2}k^{9}
\gamma ^{2}w-7020a_{3}a_{7}k^{13}\\
&-624a_{3}a_{7}k^{11}\gamma ^{2}+2466a_{3}k^{14}\gamma +190a_{3}k^{12}\gamma ^{3}-918a_{3}k^{12}w-1944a_{7}
k^{14}\\
&+639k^{15}\gamma .
\end{align*}

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\end{document}