\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 189, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/189\hfil An Abel type cubic system]
{An Abel type cubic system}

\author[G. R. Nicklason \hfil EJDE-2015/189\hfilneg]
{Gary R. Nicklason}

\address{Gary R. Nicklason \newline
Mathematics, Physics and Geology,
Cape Breton University,
Sydney, Nova Scotia, B1P 6L2, Canada}
\email{gary\_nicklason@cbu.ca}

\thanks{Submitted March 30, 2015. Published July 16, 2015.}
\subjclass[2010]{34A05, 34C25}
\keywords{Center-focus problem; Abel differential equation; constant invariant;
\hfill\break\indent symmetric centers}

\begin{abstract}
 We consider center conditions for plane polynomial systems of Abel type
 consisting of a linear center perturbed by the sum of 2 homogeneous
 polynomials of degrees $n$ and $2n-1$ where $n \ge 2$.
 Using properties of Abel equations we obtain two general systems valid
 for arbitrary values on $n$. For the cubic $n=2$ systems we find several
 sets of new center conditions, some of which show that the results in a
 paper by Hill, Lloyd and Pearson which were conjectured to be complete are
 in fact not complete. We also present a particular system which appears to
 be a counterexample to a conjecture by Zo\l\c{a}dek et al. regarding
 rational reversibility in cubic polynomial systems.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks


\section{Introduction}

In this work we consider center conditions for the origin for differential
polynomial systems in the plane having the form of a linear
center perturbed by homogeneous polynomials of degrees $n$ and $2n-1$ where
$n \ge 2$. We refer to these as \emph{generalized cubic systems} since they
contain the cubic system ($n=2$) as a particular case.
Specifically, we assume nonlinearities such that the resulting phase plane
equation is one of Abel type. The most general form of this type of system is
\begin{equation}
\frac{dx}{dt} = -y - p_1(x,y) - p_2(x,y), \quad
\frac{dy}{dt} = x + q_1(x,y) + q_2(x,y) \label{e1.1}
\end{equation}
where
\begin{equation}
\begin{gathered}
p_1(x,y) = a_0x^n + a_1x^{n-1}y, \\
p_2(x,y) = c_0x^{2 n-1} + c_1x^{2 n-2}y, \\
q_1(x,y) = b_0x^n + b_1x^{n-1}y + b_2x^{n-2}y^2 + b_3x^{n-3}y^3, \\
q_2(x,y) = d_0x^{2 n-1} + d_1x^{2 n-2}y + d_2x^{2 n-3}y^2 + d_3x^{2 n-4}y^3.
\end{gathered} \label{e1.2}
\end{equation}
where $b_3=0 $ for the cubic system. To avoid the large number of possible
subcases which can arise in \eqref{e1.2}, we assume throughout that
$a_0=c_0=0$ (except for section 6) and that all other parameters are nonzero.
Systems of a more general type consisting of a linear center
perturbed by polynomials of degrees $n$ and $2n-1$ have been studied previously.
For specific examples of this see the papers
by Gin\'e and Llibre \cite{g1,g2,g3}.

The ordinary differential equation
\begin{equation}
\frac{dy}{dx} = -\frac{x + q_1(x,y) + q_2(x,y)}{y + p_1(x,y) + p_2(x,y)}
\label{e1.3}
\end{equation}
corresponding to this system is an Abel equation of the second kind and the
properties of these equations can be used to investigate both
the integrability of the equation and the center properties of the associated
system. Most of the results which we give in this
paper are for the cubic system, but we will also give a general set of center
conditions valid for arbitrary values of $n$. For the purposes
of this work an integrating factor $\mu(x,y)$ of \eqref{e1.3} is a function
such that
$$
\frac{\partial}{\partial y} \big(\mu(x,y)(x + q_1(x,y) + q_2(x,y)) \big)
- \frac{\partial}{\partial x} \big(\mu(x,y)(y + p_1(x,y) + p_2(x,y)) \big) = 0.
$$
We shall frequently make use of the fact given by Reeb \cite{r1} that if
$\mu(x,y)$ is nonzero on a neighborhood of the critical point $(0,0)$ then
the corresponding system is a center of \eqref{e1.1},\,\eqref{e1.2}.
Every integrating factor that we determine in this paper will be of this type,
although we might not specifically mention it.


Particular cases of the cubic systems of the type defined by \eqref{e1.1},
\eqref{e1.2} have been studied extensively in the past. One of the
earliest of these was by Kukles \cite{k2} who considered the case
$a_0=a_1=c_0=c_1=0$. He proposed a set of conditions which he thought
to be both necessary and sufficient for centers of these systems.
These conditions were shown to be not complete by Jin and Wang \cite{j1}
who obtained a system not contained in the Kukles' condition.
The complete set of conditions was given in papers by Lloyd et al.
\cite{l1,p1}.

The case $a_0=c_0=c_1=0$ was considered by Hill, Lloyd and Pearson in \cite{h1}.
They obtained three separate conditions for nonzero parameters and
conjectured that this gave a complete classification of the centers for such systems.
One of the main purposes of this paper is to show that
these conditions are not complete. Using properties of Abel equations, we are able
to demonstrate another large class of centers not contained
in those given in \cite{h1}. This approach also allows us to make further statements
regarding the invariant curves and integrating factors for the systems
considered therein. The results for the cases considered previously were obtained
by calculating the Lyapunov coefficients of the system and
then using resultants to obtain common factors. This type of development often
leads to massive expressions which frequently become intractable.
Fortunately, for these Abel type systems there is another approach which will
reproduce the same results much more quickly and also allows for
extension to the case $c_1 \ne 0$. Analyzing the problem in terms of Lyapunov
coefficients allows for a discussion of both the necessity and
sufficiency of the conditions so obtained, as well as a possible determination of
the number of limit cycles which can bifurcate from the critical
point. However since the results in this paper determine only the integrability of
the systems, we consider just the sufficiency of the conditions.


In the next two sections we develop some of the fundamental ideas concerning
Abel differential equations and use them to generate the systems of
equations that define the solutions that we seek. In section 4 we employ the
methodology previously introduced to consider certain aspects of
the results given in \cite{h1}.
The case $c_1 \ne 0$ is considered in the next section and we present several
new sets of center conditions for these cubic systems. In the final
section we look at certain results by Cherkas and Romanovski \cite{c2} which cover
some of the cases defined by $a_0c_0 \ne 0$ which are not dealt with
in this paper. We present a particular system which seems to be a counterexample
to a conjecture made by \.Zo\l \c adek and others regarding rational (algebraic)
reversibility in cubic polynomial systems.


\section{Elements of the Abel differential equation}

The emphasis of this paper is to use properties of the Abel differential
equation to characterize center conditions for the system \eqref{e1.1}, \eqref{e1.2}.
The primary reason for doing this is because all of the results having nonzero
parameters in \cite{l1,p1} can be expressed in terms of easily identifiable
forms of Abel equations. These are either of \emph{Bernoulli} or
\emph{constant invariant} type, terms which we will define more fully below.
In order to do so, we briefly review certain standard aspects of Abel
equations as well as some less known ones.

An Abel equation of the first kind is given by
\begin{equation}
\frac{dy}{dx} = f_3(x)y^3 + f_2(x)y^2 + f_1(x)y + f_0(x) \label{e2.1}
\end{equation}
and an Abel equation of the second kind by
\begin{equation}
\frac{dy}{dx} = \frac{f_3(x)y^3 + f_2(x)y^2 + f_1(x)y + f_0(x)}{g_1(x)y + g_0(x)}
\label{e2.2}
\end{equation}
where the coefficient functions are assumed to be suitably differentiable
functions of $x$. The form \eqref{e2.2} can always be transformed
to an Abel equation of the first kind by the variable change
$$
y(x) = \frac{1}{g_1(x)u(x)} - \frac{g_0(x)}{g_1(x)}.
$$

If we transform \eqref{e2.1} by $y(x)=u(x) - f_2(x)/(3 f_3(x))$
we obtain the equation
\begin{equation}
\frac{du}{dx} = f_3(x)u^3 + \frac{3 f_1(x)f_3(x) - f_2^2(x)}{3 f_3(x)}u
+ \frac{s_3(x)}{f_3^2(x)} \label{e2.3}
\end{equation}
where
\begin{equation}
s_3(x) = f_0(x)f_3^2(x) + \frac{2}{27}f_2^3(x) + \frac{1}{3}
\left( f_3(x)f_2'(x) - f_2(x)f_3'(x) - f_1(x)f_2(x)f_3(x) \right). \label{e2.4}
\end{equation}
Transformations of this type have been given in Kamke \cite{k1}.
From $s_3(x)$ it is possible to define recursively an infinite sequence
of relative invariants \cite{c1} of weight $2k+1$ by
\begin{equation}
s_{2k+1}(x) = f_3(x) s_{2k-1}'(x) + (2k-1)
\Big(\frac{1}{3}f_2^2(x) - f_3'(x) - f_1(x)f_3(x) \Big)s_{2k-1}(x) \label{e2.5}
\end{equation}
for $k \ge 2$ and from these, a sequence of absolute invariants can be formed.
If the first invariant $I_1(x)=s_5^3(x)/s_3^5(x)$ is constant,
the Abel equation can be transformed to a separable equation. We shall refer
to centers defined by this condition as
\emph{constant invariant centers}.
In \cite{g1} the authors use the result
$$
f_2(x)\Big(\frac{f_3(x)}{f_2(x)}\Big)' + f_1(x)f_3(x) = Cf_2(x)^2
$$
to investigate integrability conditions for centers and foci in certain generalized
cubic systems with $n=2,\,3$. In this $C$ is a constant
and if this relation is satisfied it is sufficient to guarantee that the form
of \eqref{e2.1} with $f_0(x)=0$ has constant invariant. This is an easily derivable
extension of the more usual form of the condition (see Murphy \cite{m1})
$ \big(f_3(x)/f_2(x)\big)' = Cf_2(x) $ for which $f_1(x) = 0$ as well.
These conditions can be difficult to use with the full form \eqref{e2.1}
(if we wish to find conditions for which the equation has constant invariant)
because it is necessary to know a particular solution in order to eliminate
the $f_0(x)$ term. We can see  from equation \eqref{e2.6} below that we can
make $f_0(x)=0$ for cubic systems ($n=2,\,b_3=0$) simply by taking $d_3 = 0$.
However, we do not do this and all  of our constant invariant results in this
paper are based on the condition $d_3 \ne 0$.

If $s_3(x)=0$ then \eqref{e2.3} is a Bernoulli equation and the corresponding
systems will also be centers. This is the case for  the two systems given 
in \cite{l1}. The invariants of an Abel equation of the second kind are defined 
to be those of the corresponding Abel equation of the  first kind.

When the Abel equation defined by \eqref{e1.2},\, \eqref{e1.3} is transformed
to an Abel equation of the first kind, the resulting equation has coefficient 
functions  defined by
\begin{equation}
\begin{gathered}
f_0(x) = \frac{b_3x^{n-3} + d_3x^{2n-4}}{1 + a_1x^{n-1} + c_1x^{2n-2}}, \quad
f_1(x) = \frac{b_2x^{n-2} + d_2x^{2n-3}}{1 + a_1x^{n-1} + c_1x^{2n-2}}, \\
f_2(x) = \frac{b_1x^{n-1} + d_1x^{2n-2}}{1 + a_1x^{n-1} + c_1x^{2n-2}}, \quad
f_3(x) = \frac{x + b_0x^{n} + d_0x^{2n-1}}{1 + a_1x^{n-1} + c_1x^{2n-2}}.
\end{gathered} \label{e2.6}
\end{equation}
From these we can see that the relative and absolute invariants are rational
functions. Then from the definition of $I_1(x)$ we have
$s_3(x)=I_1(x)(s_3^2(x)/s_5(x))^3 $ so that if $I_1 \ne 0 $ is constant,
$s_3(x)$ must be the cube of a rational function up to some multiplicative
constant. If $I_1 = 0$ this is not necessarily the case and we will see that
these two separate conditions generally define two distinct classes of centers.

We conclude the section with the following very useful result. We are certain
that this must be known, but this writer has never seen it in print elsewhere.


\begin{lemma} \label{lem1}
A necessary and sufficient condition that the Abel differential
equation \eqref{e2.1} is of type constant invariant is that it has a particular 
solution of the form
\begin{equation}
y(x) = \eta(x) = \frac{1}{f_3(x)} \Big( K(s_3(x))^{(1/3)} - \frac{1}{3}f_2(x) \Big)
\label{e2.7}
\end{equation}
for some constant $K$.
\end{lemma}

We briefly sketch a proof of this. Using a variable transformation of the form
\begin{equation}
y(x) = \frac{1}{f_3^2(x)}\Big( s_3(x)u(x) - \frac{1}{3}f_2(x)f_3(x) \Big),
\label{e2.8}
\end{equation}
we can transform \eqref{e2.1} into
\begin{equation}
\frac{du}{dx} = \tilde{f}_3(x)u^3 + \tilde{f}_1(x)u + 1 \label{e2.9}
\end{equation}
where
\begin{equation}
\begin{gathered}
\tilde{f}_3(x) = \frac{s_3^2(x)}{f_3^3(x)}, \\
\tilde{f}_1(x) =
\frac{(6f_3'(x)-f_2^2(x)+3f_1(x)f_3(x))s_3(x)-3f_3(x)s_3'(x)}{3f_3(x)s_3(x)}.
\end{gathered} \label{e2.10}
\end{equation}
The transformation \eqref{e2.8} is of type which will retain the same equivalence
class (see \cite{c1}) as that of \eqref{e2.1} so the value of the invariant $I_1(x)$
is unchanged. For \eqref{e2.9} we obtain a much simpler functional form for $I_1(x)$.
We find that
\begin{equation}
I_1(x) = -\frac{\big(\tilde{f}_3'(x) + 3\tilde{f}_1(x)\tilde{f}_3(x) \big)^3}
{\tilde{f}_3(x)^4}. \label{e2.11}
\end{equation}
and assuming $I_1$ is constant, we can solve for $\tilde{f}_1(x)$ and write
\eqref{e2.9} in terms of $\tilde{f}_3(x)$ as
$$
\frac{du}{dx} = \tilde{f}_3(x)u^3
+ \frac{1}{3}\Big( (-I_1)^{1/3}\tilde{f}_3(x)^{1/3} -
\frac{\tilde{f}_3'(x)}{\tilde{f}_3(x)} \Big)u + 1 .
$$
We can now show that a particular solution of this equation is
$q(x)=K/(\tilde{f}_3(x))^{(1/3)} $ provided $K$ is a solution of the equation
\begin{equation}
3 K^3+(-I_1)^{(1/3)}K+3=0. \label{e2.12}
\end{equation}
Converting back to the original form \eqref{e2.1} gives \eqref{e2.7}.


The proof of sufficiency is quite straightforward. Assuming a solution of
the form \eqref{e2.9} and substituting into \eqref{e2.1},
we can show that the Abel equation can be reduced to
$$
3 K^3 - \frac{s_5(x)}{(s_3(x))^{5/3}}K + 3 = 0. 
$$
Since $K$ is a constant, so is $I_1$.

Since the constant invariant form of the Abel equation is always solvable,
it is not a complete surprise that a solution like \eqref{e2.7} should exist. 
Depending  upon the roots of \eqref{e2.12} it can define up to three invariant 
curves of the system and these can be used to construct \emph{Darboux} type 
integrating factors.
For our purposes we assume that a \emph{Darbouxian function} is a function of
the form
$$
\mathcal{D}(x,y) = \sum_{k=1}^N f_k(x,y)^{\alpha_k}
$$
for constants $\alpha_k$ and where each $f_k(x,y)$ is a polynomial which is 
a particular solution of \eqref{e1.3}. Each solution satisfies
$$
(x+q_1(x,y)+q_2(x,y))\frac{\partial f_k}{\partial y} 
- (y+p_1(x,y)+p_2(x,y))\frac{\partial f_k}{\partial x} 
= \lambda_k(x,y)f_k(x,y). 
$$
where $\lambda_k(x,y)$ is called the \emph{cofactor} of $f_k(x,y)$ and is a
polynomial of degree at most $2n-2$ for generalized cubic systems.
If an equation has a first integral of 
Darboux type  or a Liouvillian first integral then it has an integrating factor
of Darboux type (see \cite{c3}) with the possible inclusion of an exponential factor.
For the case where $I_1$ is constant we can use \eqref{e2.7} to generate the
forms for the invariant curves.

\section{Development of governing equations}

Here we develop the ideas of the previous section into a set of conditions which 
will lead to the integrable Bernoulli and constant invariant forms
of the Abel equation. This will generally produce a system of equations which is
simpler, although not necessarily fewer in number, than the usual
approach of determining center conditions in terms of Lyapunov coefficients.
We do not give the specific forms for these systems because
they are very straightforward to obtain. They are determined by calculating the
numerators of the appropriate expressions and forming the systems
by setting the coefficients of each power of $x$ equal to zero simultaneously.
In no case that we give did the solutions of the system result
in the vanishing of the denominator. There are usually a number of ways
to proceed in order to accomplish this, but the systems so obtained are often
not equivalent with regards to solvability. As an example of the general
procedure, the Bernoulli conditions for the generalized cubic system defined
by \eqref{e1.2},\,\eqref{e1.3} (with $a_0=c_0=0)$ are obtained by calculating 
$s_3(x)$ from \eqref{e2.4}
using the coefficient functions \eqref{e2.6}. This results in a system of
6 equations which is simple enough that it may be solvable without the use of a
computer algebra system. We give these solutions at the end of the section.
All symbolic computations in this paper were carried out using Maple 13
or later versions of the software.


It was noted in \cite{h1} ($c_1=0$) that $1 + a_1x = 0$ is an invariant line 
for the solutions obtained therein. In the following we will find that a
similar consideration holds for the more general cases $c_1 \ne 0$ that we wish
to consider. For this reason it is helpful to define a new parameter by
\begin{equation}
c_1 = \frac{1}{4} \left( a_1^2 - C_1^2 \right), \label{e3.1}
\end{equation}
which allows for factorization of the denominator $1 + a_1x + c_1x^2$
in simple linear terms. We express the following systems in terms of $C_1,$ which
has the slight disadvantage that solving these systems often produces two solutions
having similar form, only one of which is needed to define the coefficient
conditions. However, it is useful because we can directly recover the solutions
in \cite{h1} by setting $C_1 = \pm a_1$ in our results.

For the constant invariant form we consider only the cubic $n=2$ case and for
this we must distinguish between the two cases $I_1 \ne 0$ and $I_1 = 0$.
We begin by assuming that $I_1 \ne 0$. We need to obtain the functional forms for
$s_3(x)$ and $s_5(x)$ and a little consideration will show us that
they must have similar functional components in order to obtain the cancellation
necessary to produce a constant value for $I_1$. Using \eqref{e2.4}, \eqref{e2.8}
with $n=2$ and $b_3=0$ gives
\begin{equation}
s_3(x) = \frac{k(x)}{\big((a_1+C_1)x +2 \big)^3 \big((a_1-C_1)x +2 \big)^3 }
\label{e3.2}
\end{equation}
where $k$ is a polynomial in $x$ having maximum degree 6 and minimum degree 2.
(So $s_3(x)=0$ would produce 5 equations in this case.) We were able to use
this form to produce a one parameter family of solutions for the case
$c_1 \ne 0$ by taking specific values for several parameters in conjunction with
some of the following ideas. What we found for these solutions was that $s_3(x)$
actually has a much simpler form in the case when $I_1$ is constant. With this
in mind we define
\begin{equation}
\psi(x) = \frac{x}{(a_1+C_1)x +2 } \label{e3.3}
\end{equation}
and then take
\begin{equation}
s_3(x) = K_1 \psi^3(x), \quad s_5(x) = K_2 \psi^5(x) \label{e3.4}
\end{equation}
where $K_1, K_2$ are nonzero constants. These forms now give the desired constant
invariant value $I_1=K_2^3/K_1^5$.
If $C_1=a_1$ ($c_1=0$), \eqref{e3.3} reduces to the form of $s_3(x)$ for the
constant invariant solutions in \cite{h1}. We can also express
$\psi(x)$ in terms of the other linear factor of \eqref{e3.2} (i.e. $C_1 \to -C_1$) but this just leads to the duality of the solutions mentioned earlier and
produces nothing new. So one set of conditions is defined by the equivalence
of \eqref{e3.2} with \eqref{e3.3} and \eqref{e3.4}. This can be expressed as
\begin{equation}
k(x) - K_1 x^3\left((a_1-C_1)x +2 \right)^3 = 0 \label{e3.5}
\end{equation}
where the coefficients of $k(x)$ are given in terms of the parameters of the
system \eqref{e1.2}. This yields a system with 5 equations and in this we can
see that $K_1$ occurs with degree at most one in any equation. So it is easily
solved for and this underscores one of the advantages of this method.
Using the previous results, we will introduce a second independent parameter which
also occurs at low degree. Not only does this allow us to solve for these
extra parameters in terms of the system parameters, but it also seems to produce
the system which is the simplest to solve.

In addition to \eqref{e3.5} which guarantees that $s_3(x)$ has the correct form,
we need another condition to ensure that $I_1$ is constant. We have tried to
use the form of $s_5(x)$ given in \eqref{e3.4} in conjunction with the general
form from \eqref{e2.5}, but the implementation of this is sufficiently complex that
we were unable to obtain any general results using it. We can obtain a solvable
system by substituting for $s_3(x),\, s_5(x)$ from \eqref{e3.4} into the definition
of $s_5(x)$ from \eqref{e2.5}. This method gives the values of $K_1, K_2$ from
which we could calculate $I_1,$ but since this is not really of interest we
prefer to proceed in a somewhat different direction. We will make use of some of
the results arising in the proof of the Lemma in the previous section,
and in this regard the simplified functional form of $I_1(x)$ given by
\eqref{e2.11} is quite useful. Also, the final form can be easily be modified to 
cover  the remaining case $I_1=0$.

From the definition of $I_1(x)$ we see that
$s_3^2(x) = \left(s_5(x)/s_3(x) \right)^3/I_1(x)$. 
Then from \eqref{e2.10} we have
$$
\tilde{f}_3(x) = \frac{1}{I_1(x)} \Big( \frac{s_5(x)}{f_3(x)s_3(x)} \Big)^3. 
$$
Assuming $I_1$ is constant, \eqref{e2.11} gives
$$
(\tilde{f}_3(x) I_1)^{1/3} 
= -\frac{\tilde{f}_3'(x)}{\tilde{f}_3(x)} - 3 \tilde{f}_1(x) 
= \frac{s_5(x)}{f_3(x)s_3(x)}
= \frac{K_2}{K_1} \frac{\psi^2(x)}{f_3(x)}
$$
when \eqref{e3.4} is used. With \eqref{e2.10}, \eqref{e3.4} this can be written as
\begin{equation}
3 \frac{f_3'(x)}{f_3(x)} - 3\frac{\psi'(x)}{\psi(x)} - \frac{f_2^2(x)}{f_3(x)}
+ 3 f_1(x) + K'\frac{\psi^2(x)}{f_3(x)} = 0\,, \label{e3.6}
\end{equation}
where $K' = K_2/K_1$. This gives a system having 5 equations, so when combined
with \eqref{e3.5} it produces a system having 10 equations in total.


The final integrable form that we consider occurs when $I_1=0$
(i.e. $s_3(x) \ne 0$, $s_5(x)=0$), but in this case there is no reason to expect 
that $s_3(x)$  will appear as an exact cube. Fortunately, due to some previous 
work which we will describe a bit more fully in the next section, we knew that an
appropriate form is $s_3(x) = K_1 \omega(x)$ where $K_1$ is a nonzero constant and
\begin{equation}
\omega(x) = \frac{x^3}{\big((a_1+C_1)x +2 \big) \big((a_1-C_1)x +2 \big)^2 }.
\label{e3.7}
\end{equation}
As in the previous case $I_1 \ne 0$ there is an alternative form of this in
which the powers on the terms in the denominator are reversed, but this
again produces nothing new. The condition corresponding to \eqref{e3.5} becomes
\begin{equation}
k(x) - K_1 x^3 \left((a_1-C_1)x +2 \right) \left((a_1+C_1)x +2 \right)^2 = 0
\label{e3.8}
\end{equation}
and \eqref{e3.6} is replaced by
\begin{equation}
3\frac{f_3'(x)}{f_3(x)} - \frac{\omega'(x)}{\omega(x)}
- \frac{f_2^2(x)}{f_3(x)} + 3 f_1(x) = 0\,. \label{e3.9}
\end{equation}
To conclude the section we give the parameter dependencies for the Bernoulli
forms of the generalized cubic system.


\begin{proposition} \label{prop1} 
Let $ n \ge 2$ be an integer, $a_0=c_0=0$ and \eqref{e1.1}, \eqref{e1.3} 
be given by \eqref{e1.2}. Then if either
\begin{equation}
\begin{gathered}
b_0 = \frac{b_1^2d_0 + d_1^2}{b_1d_1}, \quad
b_3 = -\frac{1}{3}(n-2)b_1, \quad
c_1 = \frac{b_1d_0 (a_1d_1 + b_1d_0)}{d_1^2}
+ \frac{2}{9}\frac{d_1^2 - b_1^2d_0}{(n-1)d_0}, \\
d_2 = (n-1)\frac{b_1d_0(2b_1d_0 - a_1d_1)}{d_1^2}
+ \frac{1}{9}\frac{9b_1b_2d_0^2 + 2d_1^3}{d_0d_1}
+ \frac{2}{9}\frac{b_1^2d_0 - d_1^2}{(n-1)d_0}, \\
d_3 = \frac{1}{3}(n-1)\frac{b_1(2b_1d_0 - a_1d_1)}{d_1}
+ \frac{1}{3}\frac{b_1(a_1d_1 + b_2d_1 - b_1d_0)}{d_1}
\end{gathered}\label{e3.10}
\end{equation}
or
\begin{equation}
\begin{gathered}
a_1 = b_0 + \frac{2}{9}\frac{b_0d_1^2 -4b_1d_0d_1
+ b_0b_1^2d_0}{(n-1)d_0(b_0^2 - 4d_0)}, \quad
b_3 = -\frac{1}{3}(n-2)b_1, \\
b_2 = -a_1 - (n-2)b_0 + \frac{2}{9}\frac{2b_0d_1^2
+ b_0b_1^2d_0 - 6b_1d_0d_1}{d_0(b_0^2 - 4d_0)}, \\
c_1 = d_0 + \frac{2}{9}\frac{2b_1^2d_0^2 + b_0^2d_1^2 -2b_0b_1d_0d_1
- 2d_0d_1^2}{(n-1)d_0(b_0^2 - 4d_0)}, \\
d_2 = -c_1 - (2 n-3)d_0 + \frac{2}{9}\frac{b_0^2d_1^2 + b_0b_1d_0d_1
- 4d_0d_1^2 - 2b_1^2d_0^2}{d_0(b_0^2 - 4d_0)}, \\
d_3 = -\frac{1}{3}(2 n-3)d_1 + \frac{1}{27}\frac{(2b_0b_1d_1
-4b_1^2d_0)d_1}{d_0(b_0^2 - 4d_0)}
\end{gathered}  \label{e3.11}
\end{equation}
the corresponding systems satisfy $s_3(x)=0$ and are centers of \eqref{e1.1}.
\end{proposition}


Since these conditions are derived from the consideration of Bernoulli forms 
for \eqref{e1.3} it is obvious that integrating factors can be readily obtained.
We can show that an integrating factor for \eqref{e3.10} is given by
\begin{equation}
\mu(x,y) = \frac{e^{\int \mathcal{F}(x)\,dx}}{(3b_1d_0x^{n-1} + b_1d_1x^{n-2}y
+ 3d_1)^3  \mathcal{D}(x)} \label{e3.12}
\end{equation}
where $\mathcal{F}(x) = \mathcal{N}(x)/\mathcal{D}(x) $ with
\begin{align*}
\mathcal{N}(x)
&= x^{n-2} \big[ (9(n-1)^2b_1d_0^2(a_1d_1 + b_1d_0) + 2(n-1)d_1(2d_1^3
+ 9b_1b_2d_0^2 - 3b_1^2d_0d_1) \\
&\quad +4d_1^2(b_1^2d_0 - d_1^2))x^{n-1} + 27(n-1)^2b_1d_0^2d_1
+ 18(n-1)b_2d_0d_1^2 \big]
\end{align*}
and
\begin{align*}
\mathcal{D}(x)
&= [9(n-1)b_1d_0^2(a_1d_1 - b_1d_0) + 2d_1^2(d_1^2 - b_1^2d_0)]x^{2n-2} \\
&\quad + 9(n-1)a_1d_0d_1^2 x^{n-1} + 9(n-1)d_0d_1^2.
\end{align*}
We observe that \eqref{e3.12} is analytic and nonzero on a neighborhood of
the critical point $(0,0)$ with the same being true for the integrating
factor for \eqref{e3.11}.


\section{The article by Hill, Lloyd and Pearson}


In \cite{c4} Christopher and Lloyd consider a particular case of centers
for the case $a_1=c_1=0$ and Lloyd and Pearson look at the general case 
in \cite{l1}.
They find two general sets of centers which correspond to the Bernoulli
centers for cubic systems from Proposition \ref{prop1}. Unfortunately,
the parameter assignments for the second case in \cite{l1} are not correctly given,
but correct relations for this are found in \cite{p1}.
Hill, Lloyd and Pearson \cite{h1} consider the case $c_1=0$ and obtain three
independent sets of coefficients for centers. Upon analysis,
we once again find that two of these are Bernoulli centers and the third is
constant invariant. Since the immediate integrability of the
systems so obtained is not recognized, they undertake an exhaustive analysis to
determine invariant curves and integrating factors. They also
conjecture that the set of conditions which they obtain are complete.
All of the results in these papers were obtained by calculating and
reducing the Lyapunov coefficients of the system. In the following we will
discuss various aspects of the invariant curves and integrating factors
with regard to the methodology introduced in this paper and further, use
it to show that the conditions in \cite{h1} are not complete. There it
was conjectured \cite[Conjecture 4.2]{h1} that the results given in that paper
for systems satisfying $a_1b_1d_3 \ne 0$ are complete.


We begin by considering the question of completeness just mentioned.
Some time ago this writer undertook an analysis of the cubic system ($a_1c_1 \ne 0$)
considered in this paper. This was done by the attempted reduction of
Lyapunov coefficients along the lines used by other authors.
However, the general problem of insufficient memory quickly appeared and we
were forced to make some assumptions in order to continue.
It was decided to impose the condition $b_0 + b_2 = 0$ and this drastically
reduced the size of the expressions obtained. Even so,
it was necessary at one point to obtain the factorization of a resultant containing
more than 144,000 terms. Fortunately, this was (indirectly)
possible and through it we were able to identify a class of centers which had
not been previously reported. It turned out that they were of the
type $I_1 = 0$ arising from \eqref{e3.8},\,\eqref{e3.9} and for future reference
we denote this system by $\mathcal{S}$. These solutions were also very helpful
in enabling us to set the form of $s_3(x)$ given by \eqref{e3.7}.
So, when we became aware of the paper \cite{h1} we used these ideas to check for the
possible existence of such solutions in that system. We quickly found another
set of conditions which is given in the following.


\begin{proposition} \label{prop2} 
Let $n = 2$, $a_0=c_0=c_1=b_3=0$ and \eqref{e1.1}, \eqref{e1.3} 
be given by \eqref{e1.2}. Set $\alpha^2 = b_0/(b_0 + 2b_2)$. 
Then the system defined by
\begin{gather*}
a_1 = 3(b_0 + b_2), \quad b_1 = 2\alpha (2b_0 + 3b_2), \quad 
d_0 = \frac{1}{3}\frac{b_0^2(2b_0 + 3b_2)}{b_0 + 2b_2}, \\
d_1 = b_0 \alpha (2b_0 + 3b_2), \quad d_2 = b_0(2b_0 + 3b_2), \quad 
d_3 = \frac{\alpha}{3}(b_0 + 2b_2)(2b_0 + 3b_2)
\end{gather*}
satisfies the condition $I_1=0$ and is a center of \eqref{e1.1}.
\end{proposition}


Since $s_3(x)$ for this system is nonzero, the system is not Bernoulli. 
It could be a member of the  constant invariant family given in the paper with 
an added condition to make $I_1=0,$ but it is easily shown by a number of 
different methods  (such as the form of $s_3(x)$) that this is not the case. 
This would have required six parameter assignments such as the results
in Proposition \ref{prop2}, but the authors only consider the possibility of up to 
five such assignments. They also state that there were several cases that
could not be fully reduced.

An integrating factor $\mu(x,y)$ for the system satisfying $\mu(0,0) \ne 0$
can be constructed with the help of \eqref{e2.7}. The phase plane equation 
\eqref{e1.3}  is an Abel equation of the second kind which can be recovered 
from its first kind form by letting $ y \to 1/y. $ Then since $K=-1$ is the only
real root of \eqref{e2.12} in the case $I_1=0$, an appropriate form for the
particular solution is
\begin{equation}
\Big( \frac{f_3(x)}{y} + \frac{1}{3}f_2(x) \Big)^3 + s_3(x) = 0 \label{e4.1}
\end{equation}
where $s_3(x)$ has the form \eqref{e3.7} in which one of the conditions
$C_1 = \pm a_1$ has been imposed. The integrating factor is of Darboux type and so
its construction will involve determining the invariant curves of the system.
From \eqref{e4.1} we can identify  two such curves. One is a polynomial
$\mathcal{P}(x,y)$ of degree 4 and the other is an invariant line.
The integrating factor is then given as
$\mu(x,y) = (1 + 3(b_0+b_2)x)^{-1/3}/\mathcal{P}(x,y)$.

We now briefly consider some other aspects of the results in \cite[Theorem 3.1]{h1}, 
particularly with regards to integrating factors and invariant curves.
Specifically, parts (i) and (iii) are Bernoulli and part (ii) is constant invariant.
In their discussion the authors state that the integrating factor for case (iii) 
is the most complicated that they had encountered
and also that each system required an exponential term. As the Bernoulli forms of
these systems are directly obtainable by setting $n=2$
and letting $c_1=0$ in our results \eqref{e3.10},\eqref{e3.11}, we can see that
an integrating factor for these will include an exponential function. The one
for case (i) can be found as a particular case of \eqref{e3.12} and the other can
be obtained as a particular case of \eqref{e5.1} in the next section.
With these forms known, we now consider the remaining case.


Case (ii) produces a constant invariant equation having rational coefficients and
in our study of such equations ($I_1 \ne 0$) for polynomial systems we have always 
found that an integrating factor could be obtained in the form of a rational function.
This is again of Darboux type so we must determine invariant curves for the
system. In this regard it is straightforward to verify that the expression given 
in \cite{h1} for this case is not such a curve.
It is possible to find the general forms for these and from them to construct 
an integrating factor. This requires an explicit
description of the system and in terms of the notation in this paper
this can be given by
\begin{equation}
\begin{gathered}
a_1 = b_0 + b_2, \\
d_0 = \frac{(b_0+b_2)(b_1(b_0+b_2)\alpha + (b_0-b_2)b_1^2
 +4b_0b_2(b_0+3b_2))}{2(b_1^2+4b_2^2)}, \\
d_1 = \frac{(b_0+b_2)(3b_2(b_0+b_2)\alpha + b_1(b_1^2
 +b_2^2-3b_0^2-6b_0b_2))}{b_1^2+4b_2^2}, \\
d_2 = -\frac{(b_0+b_2)(3b_1(b_0+b_2)\alpha + 12b_0^2b_2 - b_0b_1^2 + 20b_0b_2^2
- 3b_1^2b_2)}{2(b_1^2+4b_2^2)}, \\
d_3 = \frac{(b_0+b_2)^2(b_1(b_0+b_2) - b_2\alpha )}{b_1^2+4b_2^2}
\end{gathered}\label{e4.2}
\end{equation}
where $ \alpha^2 = b_1^2 - 4b_0^2 - 8b_0b_2$. We can now use this representation
along with \eqref{e2.7} to determine information regarding invariant curves
of the system. Similar to \eqref{e4.1} the appropriate form for the particular
solution is
$$
\frac{f_3(x)}{y} + \frac{1}{3}f_2(x) - K s_3(x)^{1/3}
= \frac{f_3(x)}{y} + \frac{1}{3}f_2(x) - \frac{K' x}{a_1x+1} = 0
$$
for a constant $K'$. Of course $K'$ is known but its form is not easy to deal with.
 As a component of this solution we can readily identify
the hyperbola $ 1 + b_0 x + A(K')y + Bx^2 + C(K')xy = 0$ where $B$ is known and
$A,C$ depend upon $K'$. With this it is not difficult to determine the
invariant curve as
\begin{align*}
\mathcal{C}(x,y)
& = 1 + b_0x + \frac{1}{2}(b_1 + \alpha)y \\
&\quad + \frac{(b_0+b_2)(b_0b_1^2 + 4b_0^2b_2 + 12b_0b_2^2 - b_1^2b_2
 + b_1(b_0+b_2)\alpha)}{2(b_1^2+4b_2^2)}x^2 \\
&\quad + \frac{(b_0+b_2)(b_1^3 + 2b_1b_2^2 - 2b_0^2b_1 - 4b_0b_1b_2
 + (2b_0b_2 + b_1^2 + 6b_2^2)\alpha)}{2(b_1^2+4b_2^2)}xy
\end{align*}
In addition to this the system also has three invariant lines, one of which
is $a_1x +1 = 0$. The other two have the form $Ax + By + C = 0$ and can be
obtained by assuming that they are solutions of \eqref{e1.3}.
Letting $\ell_k(x,y) = A_kx+B_ky+C_k$ for $k=1, 2,$ we find
\begin{gather*}
A_1 = (b_0+b_2)((b_1+\alpha)\beta + b_0(4b_0^2 + 8b_0b_2 -b_1^2 - b_1\alpha)), \\
B_1 = 2b_0(b_0+b_2)^2(b_1+\alpha), \\
C_1 = (b_1+\alpha)\beta + (b_0+2b_2)(b_1^2-4b_0b_2+b_1\alpha), \\
A_2 = A_1 - 2(b_0+b_2)(b_1+\alpha)\beta, \\
 B_2=B_1, \quad C_2 = C_1 - 2(b_1+\alpha)\beta
\end{gather*}
where
\[
\beta = \sqrt{b_0(12b_0b_2^2 + 8b_0^2b_2 - b_0b_1^2 - 2b_1^2b_2
- 2b_0b_1(b_0+b_2)\alpha)}.
\]
The lines $\ell_1(x,y)$ and $\ell_2(x,y)$ will be real if $I_1 > 729/4$
and complex conjugates if $I_1 < 729/4$. In the latter case their product
will have the form of an empty conic. It can then be confirmed with the help of a
computer algebra system that the rational function
\begin{equation}
\mu(x,y) = \frac{1 + (b_0 + b_2)x}{\ell_1(x,y) \ell_2(x,y) \mathcal{C}(x,y)}
\label{e4.3}
\end{equation}
is an integrating factor of the system defined by \eqref{e4.2}.

\section{The case $c_1 \ne 0$}


 Here we extend the ideas presented in sections 2 and 3 to the system 
in which $c_1 \ne 0$. We know that there are two sets of Bernoulli
centers given as follows.


\begin{corollary} \label{coro1}
Let $n=2$, $a_0=c_0=0$ and \eqref{e1.1}, \eqref{e1.3} be given by \eqref{e1.2}. 
Then if either
\begin{gather*}
b_0 = \frac{b_1^2d_0 + d_1^2}{b_1d_1}, \quad b_3 = 0, \quad
c_1 = \frac{9a_1b_1d_0^2d_1 - 2b_1^2d_0d_1^2 - 9b_1^2d_0^3 + 2d_1^4}{9d_0d_1^2} , \\
d_2 = \frac{b_1(9b_2d_0d_1 -9a_1d_0d_1 + 2b_1d_1^2 +18b_1d_0^2)}{9d_1^2} , \quad 
d_3 = \frac{b_1(b_1d_0 + b_2d_1)}{3d_1}
\end{gather*}
or
\begin{gather*}
a_1 = b_0 + \frac{2}{9}\frac{b_0d_1^2 -4b_1d_0d_1 + b_0b_1^2d_0}{d_0(b_0^2 - 4d_0)},
 \quad b_3 = 0, \\
b_2 = -a_1 + \frac{2}{9}\frac{2b_0d_1^2 + b_0b_1^2d_0 - 6b_1d_0d_1}{d_0(b_0^2 - 4d_0)},
 \\
c_1 = d_0 + \frac{2}{9}\frac{2b_1^2d_0^2 + b_0^2d_1^2 -2b_0b_1d_0d_1 - 2d_0d_1^2}
 {d_0(b_0^2 - 4d_0)}, \\
d_2 = -c_1 - d_0 + \frac{2}{9}\frac{b_0^2d_1^2 + b_0b_1d_0d_1 - 4d_0d_1^2 
 - 2b_1^2d_0^2}{d_0(b_0^2 - 4d_0)}, \\
d_3 = -\frac{1}{3}d_1 + \frac{1}{27}
 \frac{(2b_0b_1d_1 -4b_1^2d_0)d_1}{d_0(b_0^2 - 4d_0)}
\end{gather*}
the corresponding systems satisfy $s_3(x)=0$ and are centers of \eqref{e1.1}.
\end{corollary}

An integrating factor for the first case can be obtained from \eqref{e3.12} 
and one for the second case is given by
\begin{equation}
\mu(x,y) = \frac{e^{\int \mathcal{F}(x)\,dx}}{(3d_0x^2 + d_1xy + b_1y + 3)^3
 \mathcal{D}(x)} \label{e5.1}
\end{equation}
where $\mathcal{F}(x) = \mathcal{N}(x)/\mathcal{D}(x) $ with
\begin{align*}
\mathcal{N}(x) &= 2(3b_0^2d_1^2 + 9b_0^2d_0^2 + 4b_1^2d_0^2 - 4d_0d_1^2
- 6b_0b_1d_0d_1 - 36d_0^3)x \\
&\quad + 9b_0^3d_0 - 36b_0d_0^2 - 8b_1d_0d_1 + 4b_0d_1^2
\end{align*}
and
\begin{align*}
\mathcal{D}(x)
&= (9b_0^2d_0^2 + 2b_0^2d_1^2 -4b_0b_1d_0d_1 + 4b_1^2d_0^2 - 4d_0d_1^2
- 36d_0^3)x^2 \\
&\quad + (9b_0^3d_0 + 2b_0b_1^2d_0 + 2b_0d_1^2 - 36b_0d_0^2 - 8b_1d_0d_1)x
+ 9d_0(b_0^2 - 4d_0).
\end{align*}
 The system also has general constant invariant solutions and using the results
from section 3, we obtain the following result.


\begin{proposition} \label{prop3} 
Let $n = 2$, $a_0=c_0=b_3=0$ and \eqref{e1.1}, \eqref{e1.3} be given by 
\eqref{e1.2}. Then the system defined by
\begin{gather*}
b_0 = \frac{1}{4} \frac{\alpha}{C}, \quad b_2 = -b_0 - \frac{C}{2}, \\
b_1 = -\frac{1}{8} \frac{C^4 + 2a_1C^3 - 4d_0C^2 - 8a_1d_0C + 32d_0^2 + 32d_3^2}{d_3C}
+ \frac{1}{16}\frac{C^2 - 8d_0}{d_3C}\alpha, \\
d_1 = -\frac{1}{2}b_1C - 3d_3, \quad 
d_2 = \frac{1}{2}C^2 + a_1C - 3d_0 - \frac{\alpha}{4}
\end{gather*}
where $\alpha = 2a_1C - 8d_0 \pm 2\sqrt{C^2C_1^2 - 16d_3^2}$,
$C=C_1 - a_1$ and $C_1$ is given by \eqref{e3.1} satisfies the
condition $I_1$ constant and is a center of \eqref{e1.1}.
\end{proposition}


 The system has a rational integrating factor having the same basic form 
as \eqref{e4.3} except for the addition of a second invariant
line which appears as a factor in the denominator. This arises from the 
factorization of the denominators of the coefficient
functions $f_0,\dots,f_3$. For later reference we also specifically note that 
no member of this family satisfies the condition
$b_0 + b_2 = 0$ since $C$ cannot be zero.


 In addition to the general constant invariant solutions, the system also 
has analogs of the $I_1=0$ solution given in Proposition \ref{prop2}.


\begin{proposition} \label{prop4} 
Let $n = 2$, $a_0=c_0=b_3=0$ and \eqref{e1.1}, \eqref{e1.3} be given 
by \eqref{e1.2}. If any one of the following conditions holds, then
the corresponding system satisfies $I_1 = 0$ and is a center of \eqref{e1.1}.

\noindent (i) 
\begin{gather*}
C_1 = \alpha, \quad b_1 = \beta, \quad 
b_2 = \frac{1}{6}\alpha + \frac{1}{2}a_1 - b_0, \\
d_0 = \frac{1}{12}\alpha^2 + \frac{1}{6}(a_1 - b_0)\alpha 
 - \frac{1}{4}a_1(a_1 - 2b_0), \\
\begin{aligned}
d_1 &= \frac{1}{4}\frac{4a_1 - 5b_0}{\beta}\alpha^2
 + \frac{1}{2}\frac{(2a_1 - b_0)(2a_1 - 5b_0)}{\beta}\alpha \\
 &\quad -\frac{3}{4}\frac{4a_1^3 -15a_1^2b_0 + 26a_1b_0^2 - 16b_0^3}{\beta},
\end{aligned}\\
d_2 = \frac{3(2a_1 - 3b_0)}{b_0}d_0 - a_1\alpha + a_1^2 - 8a_1b_0 + 8b_0^2, \\
\begin{aligned}
d_3 &= -\frac{1}{12}\frac{a_1 - 2b_0}{\beta}\alpha^2 
 - \frac{1}{6}\frac{(a_1 + b_0)(a_1 - 2b_0)}{\beta}\alpha \\
 &\quad +\frac{1}{12}\frac{3a_1^3 -12a_1^2b_0 + 24a_1b_0^2 - 16b_0^3}{\beta}
\end{aligned}
\end{gather*}
where $\alpha$ is a root of the cubic in $X$,
\[
X^3 + (5a_1 - 8b_0)X^2 + 3a_1(a_1 - 4b_0)X - 9a_1^3 + 36a_1^2b_0 - 72a_1b_0^2
+ 48b_0^3 = 0,
\]
 and
\[
 \beta^2 = \frac{1}{2b_0}((3a_1 - 2b_0)\alpha^2 + (6a_1^2 -19a_1b_0 
+ 7b_0^2)\alpha - 9a_1^3 + 33a_1^2b_0 - 63a_1b_0^2 + 42b_0^3).
\]

\noindent(ii)
\begin{gather*}
b_1 = \alpha, \quad b_2 = \frac{C}{2} - b_0, \quad d_0 = -\frac{1}{4}C(C - 2b_0), \\
d_1 = -\frac{3}{4}\frac{(a_1 - b_0)C(C - 2b_0)}{\alpha}, \quad
d_2 = -d_0, \quad d_3 = \frac{1}{6}\frac{C_1(a_1 - b_0)(C - 2b_0)}{\alpha}
\end{gather*}
where $\alpha^2 = (3b_0C_1 - 3a_1C + 3b_0(3a_1 - 2b_0))/2$ and $C = C_1 + a_1$.

\noindent(iii)
\begin{gather*}
a_1 = \alpha, \quad b_1 = \beta, \quad 
b_2 = \frac{1}{2}\alpha + \frac{1}{6}C_1 - b_0, \\
d_1 = \frac{3(\alpha - b_0)d_0}{\beta} + \frac{3C_1d_0^2}{(b_0^2 - 3d_0)\beta}, \quad 
d_2 = -d_0 + \frac{2C_1d_0^2}{b_0(b_0^2 - 3d_0)}, \\
d_3 = -\frac{1}{6}\frac{b_0C_1(b_0^2 - 6d_0)\alpha}{(b_0^2 - 3d_0)\beta} 
 + \frac{1}{6}\frac{b_0C_1^2(b_0^2 - 2d_0)}{(b_0^2 - 3d_0)\beta}
+ \frac{1}{3}\frac{b_0^2C_1(b_0^2 - 5d_0)}{(b_0^2 - 3d_0)\beta}
\end{gather*} 
where $\alpha$ is a root of the quadratic in $X$,
\begin{align*}
&b_0(b_0^2 - 3d_0)X^2 - 2b_0^2(b_0^2 - 3d_0)X - b_0(b_0^2 - 3d_0)C_1^2 
-2(b_0^2 - 4d_0)(b_0^2 - d_0)C_1\\
& + 4b_0d_0(b_0^2 - 3d_0) = 0,
\end{align*}
 and
$$ 
\beta^2 = \frac{3}{2}b_0\alpha - \frac{3}{2}\frac{b_0C_1(b_0^2- 5d_0)}{b_0^2 
- 3d_0} - 3b_0^2 + 6d_0. 
$$

\noindent (iv)
\begin{gather*}
a_1 = \alpha, \quad b_1 = \beta, \quad 
b_2 = \frac{1}{2}\alpha + \frac{1}{6}C_1 - b_0, \\
d_0 = \frac{1}{12}(\alpha + C_1)(3\alpha - C_1), \\
\begin{aligned}
d_1 &= -\frac{1}{2}\frac{C_1 + 3b_0}{\beta}\alpha^2 - \frac{1}{6}\frac{2C_1^3 
 + 9b_0C_1^2 - 9b_0^2C_1 - 45b_0^3}{(C_1 + 3b_0)\beta}\alpha \\
&\quad +\frac{1}{6}\frac{C_1^4 + 9b_0C_1^3 + 24b_0^2C_1^2 + 15b_0^3C_1 
 - 18b_0^4}{(C_1 + 3b_0)\beta}, 
\end{aligned} \\
d_2 = \frac{1}{12}\frac{(\alpha + C_1)(2C_1^2 + 9b_0C_1 - 6C_1\alpha - 15b_0\alpha 
+ 12b_0^2)}{b_0}, \\
\begin{aligned}
d_3 &= -\frac{1}{6}\frac{(C_1 + 3b_0)(C_1 - 2b_0)}{b_0\beta}\alpha^2
- \frac{1}{18}\frac{2C_1^3 - b_0C_1^2 - 3b_0^2C_1 + 36b_0^3}{b_0\beta}\alpha\\
&\quad +\frac{1}{18}\frac{C_1^4 + 4b_0C_1^3 + 3b_0^2C_1^2 + 18b_0^4}{b_0\beta}
\end{aligned}
\end{gather*}
where $\alpha$ is a root of the cubic in $X$,
\begin{align*}
&9(C_1 + 3b_0)X^3 + 3(C_1^2 - 9b_0^2)X^2 - (5C_1^3 + 21\,b_0C_1^2 
+ 36b_0^2C_1 + 36b_0^3)X\\
&\quad  + C_1^4 + 6b_0C_1^3 + 15b_0^2C_1^2 + 24b_0^3C_1 + 36b_0^4 = 0,
\end{align*}
 and
$$ 
\beta^2 = -\frac{1}{2b_0}(3(C_1 + 2b_0)\alpha^2 + (2C_1^2 + b_0C_1 - 12b_0^2)\alpha 
- C_1^3 - 5b_0C_1^2 - 6b_0^2C_1 + 6b_0^3).
 $$
\end{proposition}

These are the systems which remain after removing other simpler forms which 
appear as special cases of these. At this time we believe the systems to
be independent, although it is certainly possible that it may be shown that 
one or more of them belong to a
larger, more general system. The fact that there are multiple solutions is 
somewhat surprising, although it is similar to what happens for the
$I_1 = 0$ case \cite{n1} for the homogeneous $n = 4$ system. It can be shown 
by considering resultants or numerically that each of the 4 systems
does produce real--valued parameters. Also, each of them has a form for which 
$b_0 + b_2 = 0$, although the one for (iv) is not real-valued. This is in contrast
to the result from Proposition \ref{prop3} for which the general system did not have this
 property and shows that the two classes of centers are distinct. We do not
give integrating factors for these systems because they are similar to those 
already described and can be obtained in the same manner.


 Any of the systems for which $I_1 = 0, s_3(x) \ne 0$ should be obtainable simply 
by solving the system of equations defined by $s_5(x) = 0$
(denote this system by $\mathcal{T}$) with the condition that $s_3(x)$ is nonzero. 
In fact, if this had been possible these results would have
been obtained much sooner than they actually were since these were some of the 
first solutions that we checked for following the Bernoulli cases.
However, $\mathcal{T}$ turned out to be totally intractable for the software 
(Maple) and it was not until we began adding more specific forms for
the various expressions that any results were forthcoming. Obviously, the system 
$\mathcal{S}$ described in the previous section is a solution of $\mathcal{T}$.
It was obtained through the reduction of Lyapunov constants, but even then in 
the factored forms of the resultants Maple was unable to solve the
much simplified system. Fortunately, we were able to find a parametric form for 
the solution and use it to confirm the $I_1 = 0$ nature of the system.
The system $\mathcal{S}$ is equivalent to (iii) with the added condition 
$b_0 + b_2 = 0$.

 We have indicated that we had previously attempted an analysis of this 
system $(c_1 \ne 0)$ in terms of Lyapunov coefficients. At that time we were 
unable to calculate a sufficient number of these coefficients to carry out an 
analysis without imposing additional conditions on the system parameters. 
This author is of the opinion that the results presented in this section are not 
obtainable at present using the conventional development just described because 
of the inability of computers and computer algebra systems to deal with the 
massive expressions required for this type of analysis.


\section{The case $ a_0c_0 \ne 0$}


 This case is not the general focus of this paper, but we recently learned of 
the paper \cite{c2} by Cherkas and Romanovski which addressed certain aspects of
these types of systems. They considered certain cubic systems which could be 
transformed to a system of Li\'enard type and used the known center conditions
for these systems (see \cite{c5}) to obtain center conditions for the original 
systems. Specifically, their results apply to systems of the form
$$ 
\frac{dx}{dt} = -y - a_0x^2 -a_1xy - c_0x^3, \quad 
\frac{dy}{dt} = x + b_0x^2 + b_1xy + b_2y^2 + d_0x^3 + d_1x^2y. 
$$
We have only analyzed a few of their results in detail, restricting for now 
our attention to case ($\beta$) of \cite[Theorem 5]{c2} where 41 separate systems
are listed. Each of these depends upon at least one arbitrary parameter which is 
fortunate because it is necessary to use it in order to transform the results
$(\beta)$ to cubic systems. Otherwise, the system of equations which defines this 
transformation is overdetermined and has no solution. Even so, some cases do not 
have solutions and others degenerate to quadratic systems. Of those which define 
cubic systems some are Bernoulli and others are constant invariant. 
However, there is one system defined by result 9 which is particularly interesting. 
This is given by
\begin{equation}
\begin{gathered}
\frac{dx}{dt} = -y - Ax^2 - xy - Ax^3, \\
\frac{dy}{dt} = x + x^2 + (2A-1)xy - \frac{2}{3}y^2 + 2A(1 - 5A)x^3
+ \frac{1}{3}(2A - 1)x^2y.
\end{gathered} \label{e6.1}
\end{equation}
where $A$ is an arbitrary parameter. We have found 6 integrable cases for this
system. Converting it to a first kind form, we see that only $f_3(x)$ is
dependent upon $A$ with
$$
f_3(x)=\frac{A(1 - 4A)x^4 + 9A(1 - 4A)x^3 + 3x^2 + 3x}{3(x+1)}
$$
and since the terms in $A$ are symmetric about $A=1/8$, the integrable forms
occur in pairs. Calculation of $s_5(x)$ easily shows that if $A = 1/12,1/6$
the equation is solvable with $I_1 = 0$. For $A= 1/20,\,1/5$ the systems have
a Darboux first integral which arises from a non--constant invariant Abel equation.
For $A=1/5$ this is given by
$\mathcal{U}(x,y) = \mathcal{P}^2(x,y)/\mathcal{Q}^3(x,y)$ where
\begin{equation}
\begin{gathered}
\begin{aligned}
\mathcal{P}(x,y)
&= 648x^3 - 900x^2y + 450xy^2 - 125y^3 \\
&\quad + 4500x^2 - 3060xy + 675y^2 + 7650x - 2295y + 3825,
\end{aligned}\\
\mathcal{Q}(x,y) = 84x^2 - 60xy + 25y^2 + 300x - 90y + 225.
\end{gathered} \label{e6.2}
\end{equation}
The other forms are obtained by noting that $f_3(x)$ will reduce to just
 $x$ if $A = 0,1/4$. The resulting Abel equation
$$
\frac{dy}{dx} = xy^3 - \frac{1}{3} \frac{x(x+3)}{x+1}y^2
- \frac{1}{3} \frac{2}{x+1}y
$$
(where we have retained $y$ as the dependent variable) is transformable to
a Riccati equation which is solvable in terms of special functions.
The exact form of the transformation which achieves this is difficult
to give because of its implicit nature. However, if we set
\begin{gather*}
\frac{dt}{dx} = \frac{1}{\root{3}\of{81}}\frac{x}{(x+1)^{4/3}},\quad
t = \int t'(x)\, dx = \frac{\root{3}\of{9}}{6}\frac{x+3}{(x+1)^{1/3}},\\
u(t) = \frac{1}{\root{3}\of{9}}\frac{1}{(x+1)^{2/3}}y(x)
\end{gather*}
the Abel equation transforms directly to
$$
\frac{du}{dt} = u^3 - 2t u^2
$$
which is the representative equation for Class 2 given in \cite[Appendix A]{c1}
and which is shown there to be solvable in terms of Airy functions.
These are the only cubic systems that we know of with this property,
although \.Zo\l\c{a}dek \cite[Remark 2.4]{z2} has discussed this phenomenon
as a common occurrence with regards to complex cubic systems.
No real, homogeneous ($p_2 = q_2 = 0$) system of degree 3 is solvable in such
a manner although there are homogeneous systems which are so integrable for each
$n \ge 4$. The separate cases also show that we needn't bother trying to find
an integrating factor of Darboux type for the general system even
though such expressions exist for the $I_1 = 0$ and Darboux integrable forms.

 It has been conjectured by \.Zo\l \c adek \cite{z1} that all real centers for
polynomial systems are either rationally reversible or have a Liouvillian first 
integral. Other similar conjectures have been made. See for example
Christopher and Llibre \cite{c3}. It has been shown by Berthier, Cerveau and 
Lins-Neto \cite{b1} that this is not true in general, however
the conjecture still remains for cubic systems. 
In the major work \cite{z1} and the follow-up paper \cite{z3}
the author gives a complete list of rationally reversible cubic systems. 
Minor corrections to this list are given by von
Bothmer \cite{g4} and von Bothmer and Kr\"oker \cite{g5}. 
There are 17 such systems given denoted by $CR_{j}$ for $j = 1,\dots,17$. It has
been shown by von Bothmer that $CR_5,\,CR_7,\,CR_{12}$ and $CR_{16}$ are 
fully integrable and of these $CR_5$ and $CR_7$
are constant invariant. Our brief recent study of these systems has added 
$CR_2$ and $CR_3$ as being of Bernoulli type and $CR_{17}$
is of type $I_1=0$ when expressed as $dx/dy=\dot{x}/\dot{y}$. 
In addition to these several other systems are almost integrable in the sense 
that one additional parameter assignment will lead to an integrable Liouvillian form. 
In \cite{z3} he also gives a list of 35 Darboux integrable cubic systems
which does not appear to include the system \eqref{e6.2} given above.

 The system \eqref{e6.1} has a non--Liouvillian first integral if $A=0,\,1/4$, 
so unless these two cases are rationally reversible the conjectures just mentioned
are not true. Our study of the rationally reversible cubic systems $CR_j$ has 
shown that \eqref{e6.1} is not a member of these systems. 
In particular, it cannot be a member of any of the fully integrable systems 
described above.



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\end{document}