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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 130, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/130\hfil Constant invariant solutions]
{Constant invariant solutions of the Poincar\'e center-focus problem}

\author[G. R. Nicklason\hfil EJDE-2010/130\hfilneg]
{Gary R. Nicklason}

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

\thanks{Submitted June 7, 2010. Published September 14, 2010.}
\subjclass[2000]{34A05, 34C25}
\keywords{Center-focus problem; Abel differential equation;
 constant invariant; \hfill\break\indent symmetric centers}

\begin{abstract}
 We consider the classical Poincar\'e problem
 $$
 \frac{dx}{dt}=-y-p(x,y),\quad \frac{dy}{dt}=x+q(x,y)
 $$
 where $p,q$ are homogeneous polynomials of degree $n \geq 2$.
 Associated with this system is an Abel differential equation
 $$
 \frac{d\rho}{d\theta}=\psi_3\rho^3 + \psi_2\rho^2
 $$ 
 in which the  coefficients are trigonometric polynomials.
 We investigate two separate conditions which produce a constant
 first absolute invariant of this equation. One of these conditions
 leads to a new class of integrable, center conditions for the
 Poincar\'e problem if $n \geq 9$ is an odd integer.
 We also show that both classes of solutions produce polynomial
 solutions to the problem.
\end{abstract}

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

\section{Introduction}

We consider the problem of finding center conditions for the
critical point $(0,0)$ for the system
\[
\frac{dx}{dt} = P(x,y), \quad \frac{dy}{dt} = Q(x,y)
\]
where $P$ and $Q$ are polynomials satisfying $P(0,0)=Q(0,0)=0$.
That is, we seek to find conditions on $P$ and $Q$ such that all
trajectories within a sufficiently small neighborhood of the
origin will be closed. In this work we look at the particular
system
\begin{equation}
\frac{dx}{dt} = -y - p(x,y),\quad \frac{dy}{dt} = x +
q(x,y) \label{e1}
\end{equation}
where $p$ and $q$ are homogeneous polynomials of degree $n \ge 2$.
This problem has been the focus of much research since it was
first formulated by Poincar\'e \cite{p1}.

Corresponding to \eqref{e1} is the first order differential
equation
\begin{equation}
\frac{dy}{dx} = -\frac{x + q(x,y)}{y + p(x,y)}. \label{e2}
\end{equation}
 Using a polar coordinate transformation, this is transformed into
\begin{equation}
\frac{dr}{d\theta} = \frac{\xi(\theta)\,r(\theta)^n}{1+\eta(\theta)\,
r(\theta)^{n-1}} \label{e3}
\end{equation}
where $\xi, \eta$ are trigonometric polynomials of degree $n+1$.
This can be further transformed \cite{c5} to an
Abel equation of the first kind
\begin{equation}
\frac{d\rho}{d\theta} = -(n-1)\xi(\theta)\eta(\theta) \rho^3
+((n-1)\xi(\theta) - \eta'(\theta)) \rho^2. \label{e4}
\end{equation}
 Standard center conditions for a general $n$ are given in terms
of \eqref{e3}, \eqref{e4} by
\begin{equation}
\xi(\theta)=K\eta'(\theta) \label{e5}
\end{equation}
and by $r(\theta_0-\theta) = r(\theta-\theta_0)$ (the {\it
symmetric or time reversible centers}). In these $K \ne 0$ and
$\theta_0$ are constants. In a recent paper \cite{n1} the author
was able to demonstrate a class of centers of
\eqref{e1}--\eqref{e2} by appealing to certain parity properties
of the solution of \eqref{e4}.

More generally, if \eqref{e2} has a solution expressible in the
form $U(x,y)=C$ where $U$ is analytic on a neighborhood of the
origin and $C$ is a constant, the origin will be a center. The
search for integrability conditions for \eqref{e2}, usually in the
form of an integrating factor, is one method of determining center
conditions.

In this article we consider solutions of \eqref{e3}, \eqref{e4}.
The work is motivated by the fact that if the first absolute
invariant $I_1$ of the Abel equation \eqref{e4} is constant, then
the equation is solvable. If the coefficient functions $\xi, \eta$
satisfy \eqref{e5}, it is straightforward to show that the
invariant is constant. We mention this case for completeness
purposes. Beyond this, we knew from previous work that other
constant invariant conditions for \eqref{e4} exist. In particular,
the complete symmetric case for $n=3$, which does not generally
satisfy the condition \eqref{e5}, is integrable in terms of a
constant invariant Abel equation. Also, the symmetric cases for
$n=5, 7, 9,11$, while not generally having constant invariant
$I_1$, contain subcases that are. It is our purpose here to give a
more compete description of these cases, and in so doing, give
new, integrable center conditions for \eqref{e1}--\eqref{e2}.

In the next section we review the derivation of eqs. \eqref{e3},
\eqref{e4} as well as some aspects of the Abel differential
equation. In particular, we define the relative invariants of an
Abel equation and indicate that if the first absolute invariant
$I_1$ obtained from these is constant, then the equation can be
transformed to a separable equation. We mention a particular form
for which a constant $I_1$ can always be obtained and indicate how
this would apply to the form obtained from \eqref{e3}--\eqref{e5}.
We next look at an integrating factor for \eqref{e3} and use the
previously obtained results to give a precise form for it.

In section 4 we show that we obtain a constant invariant form for
\eqref{e4} which does not satisfy \eqref{e5}. This form occurs
only when $n \ge 5$ is an odd integer. Using the integrating
factor, we obtain the solutions of \eqref{e3} and show that they
are analytic on a neighborhood of the origin. These solutions are
seen to generalize, to any odd integer, previously given solutions
for the symmetric case of $n=3$. In the next section we develop
families of polynomial solutions from each of the constant
invariant cases of \eqref{e4} discussed in this paper. In the last
section we discuss some of the reasons which led us to assume the
particular form of the integrating factor of \eqref{e3}.

\section{Aspects of the Abel differential equation}

If we set $x=r\cos \theta$, $y=r\sin \theta$ in \eqref{e2}, we
obtain the polar form \eqref{e3} where
\begin{equation}
\begin{gathered}
  \xi(\theta) = \sin\theta\, q(\cos\theta, \sin\theta) -
 \cos\theta\, p(\cos\theta, \sin\theta), \\
  \eta(\theta) = \sin\theta\, p(\cos\theta, \sin\theta) +
 \cos\theta\, q(\cos\theta, \sin\theta) \label{e6}
\end{gathered}
\end{equation}
are homogeneous trigonometric polynomials of degree $n+1$ in $\cos
\theta, \sin \theta$. Throughout the remainder of the paper we
shall refer to the degree of a homogeneous, trigonometric
polynomial as the total degree of $\cos \theta, \sin \theta$ in
each term. Following \cite{c5}, we can further transform \eqref{e3}
by setting
\[
\rho(\theta) = \frac{r(\theta)^{n-1}}{1+\eta(\theta) r(\theta)^{n-1}}.
\]
This gives \eqref{e4}
\[
\frac{d\rho}{d\theta} = -(n-1)\xi(\theta)\eta(\theta) \rho^3 +
 ((n-1)\xi(\theta) - \eta'(\theta)) \rho^2.
\]


Now that we have obtained the form of the equation \eqref{e4}
being considered, we briefly review certain properties of Abel
equations. For a general Abel equation of the first kind
$y'=g_3(x)y^3 + g_2(x)y^2 + g_1(x)y+g_0(x)$, it is possible to
define recursively an infinite sequence of relative invariants by
\cite{c4}
\begin{equation}
 s_3(x) = g_0(x)g_3^2(x) + \frac{2}{27}g_2^3(x) + \frac{1}{3}
 ( g_3(x)g_2'(x) - g_2(x)g_3'(x) - g_1(x)g_2(x)g_3(x) )
\label{e7}
\end{equation}
and
\begin{equation}
s_{2k+1}(x) = g_3(x) s_{2k-1}'(x) + (2k-1)
(\frac{1}{3}g_2^2(x) - g_3'(x) - g_1(x)g_3(x) )
s_{2k-1}(x) \label{e8}
\end{equation}
for $k \ge 2$. From these, a sequence of absolute invariants can
be formed. If the first invariant $I_1=s_5^3/s_3^5$ is constant,
the Abel equation can be transformed to a separable equation. This
is the only general class of Abel equations which is integrable by
quadrature. We note that if $s_3(x)=0,$ the Abel equation is
transformable to Bernoulli equation. The values of the parameters
for which this will occur will be obvious in what follows, so we
do not specifically consider this possibility in this paper.

 A sufficient condition that an Abel
equation of the form \eqref{e4} has a constant first invariant
$I_1$ is that the coefficient functions satisfy a relation of the
form
\begin{equation}
(\frac{g_3(x)}{g_2(x)})' = Cg_2(x) \label{e9}
\end{equation}
where $C$ is a constant. This gives
\begin{equation}
 I_1 = {\frac {729 ( 1-3C ) ^{3}}{ ( 9\,C-2 ) ^{2}}}. \label{e10}
\end{equation}
It is clearly evident that if the coefficient functions of
\eqref{e4} satisfy \eqref{e5}, the invariant is constant. It
should be noted however, that $K$ and $\eta$ in \eqref{e5} cannot
be chosen in an arbitrary fashion. We must have either
$K=-1/(n+1)$\, (the Hamiltonian condition) or $\eta$ such that
terms of the form $e^{\pm i(n+1)\theta}$ are absent. Otherwise, in
the latter case, the functions $p, q$ defined from \eqref{e6} will
not be polynomials.

The constant invariant condition associated with \eqref{e5} is
well known and the condition itself can be derived in a number of
different ways. In the following we seek a more complete
characterization of other conditions which produce a constant
invariant $I_1$ but for which \eqref{e9} is not satisfied. The
basis for carrying out this work is prior knowledge of certain
conditions of this type.

\section{An integrating factor}

The constant invariant case defined by \eqref{e5} is valid for any
integer $n \ge 2$. For the next case we require that $n \ge 5$ be
an odd integer. This is necessitated by the fact that the
expressions for $\xi, \eta$ that we shall obtain involve even
powers and this can occur only if $n$ is odd. We begin by
considering an integrating factor for the symmetric $n=3$ case
defined by
\[
p(x,y) = a_{21}x^2y + a_{03}y^3, \quad
q(x,y) = b_{30}x^3 + b_{12}xy^2.
\]
Such an expression for the polar coordinate form \eqref{e3} is
given by
\begin{equation}
\mu(r,\theta) = \frac{r}{1 + r^{2}f_1(\theta) + r^{4}f_2(\theta)}
\label{e11}
\end{equation}
where
\[
f_1(\theta) = (a_{21}-b_{12}) \cos^2 \theta +
 \frac{2a_{03}b_{30}-b_{12}a_{21}+a_{03}b_{12}
-b_{12}^2-a_{03}a_{21}}{a_{03}-a_{21}-b_{12}+b_{30}}
\]
and
\begin{align*}
f_2(\theta) 
&= (a_{03}b_{30}-a_{21}b_{12})\Big( \cos^4\theta 
+ \frac{a_{21}+b_{12}-2a_{03}}{a_{03}-a_{21}-b_{12}
 +b_{30}} \cos^2 \theta \\
&\quad + \frac{a_{03}}{a_{03}-a_{21}-b_{12}+b_{30}} \Big).
\end{align*}
In particular, we find that
\begin{equation}
 \eta(\theta) = \frac{a_{03}-a_{21}-b_{12}+b_{30}}{a_{03}b_{30}
-a_{21}b_{12}} f_2(\theta) \label{e12}
\end{equation}
 where $\eta$ is given by \eqref{e6}.

Based on earlier work involving the constant invariant cases for
$n=5, 7, 9, 11$, we modify the form given by \eqref{e11} to
\begin{equation}
\mu(r,\theta) = \frac{r^{2\alpha+1}}{1 + r^{n-1} f_1(\theta)
+ r^{2(n-1)} f_2(\theta)} \label{e13}
\end{equation}
where $\alpha$ is a value dependent on $n$ and $f_1, f_2$ are
trigonometric polynomials of degree $n-1$ and $2(n-1)$
respectively. (As we shall see, the actual degrees of $f_1, f_2$
are less than these values.) The form \eqref{e13} is not
particularly suitable for finding conditions for $f_1,f_2$ so we
set $V=1/\mu$ in the general condition for exactness
\[
\frac{\partial}{\partial r} ( \mu r^n \xi ) = 
-\frac{\partial}{\partial \theta} (\mu(1 + r^{n-1}\eta) )
\]
to obtain
\[
r^n\xi \frac{\partial V}{\partial r} + (1 + r^{n-1}\eta)
\frac{\partial V}{\partial \theta}
- r^{n-1}(n\xi + \eta') V = 0.
\]
Substituting the appropriate form of \eqref{e13} in this and
collecting powers of $r$ yields the system of equations
\begin{gather}
  (n+2\alpha+1)\xi + \eta' -f_1' = 0  \label{e14a} \\
  2(\alpha+1)\xi f_1 - f_2' - \eta f_1' + \eta'f_1 = 0 \label{e14b} \\
  (n-2\alpha-3)\xi f_2 + \eta f_2' - \eta' f_2 = 0 \label{e14c}
\end{gather}
 which the functions $\xi, \eta, f_1, f_2$ must satisfy.

In the process of solving \eqref{e14a}--\eqref{e14c},
we shall introduce two
assumptions which will produce sufficient conditions for a
solution to be found. They are both based on the known forms of
\eqref{e13} for previously obtained integrating factors. The first
of these is to assume that we can write
\begin{equation}
 \eta' = \lambda\, f_1', \quad\quad \xi = \psi\, f_1' \label{e15}
\end{equation}
where $\lambda, \psi$ are trigonometric polynomials to be
determined. The coefficient functions in \eqref{e11} and $\xi,
\eta$ for the symmetric $n=3$ case also satisfy \eqref{e15}.

The conditions \eqref{e15} may seem rather artificial, however we
have found by studying other integrating factors for integrable
forms of \eqref{e2}--\eqref{e3} that they frequently satisfy
relations of a similar type. For example, the functions
$f_1(\theta)=2a(\cos \theta)^{n-1} -2b(\sin \theta)^{n-1}$ and
$\eta$ for the system
\[
p(x,y)=ax^{n-1}y - 2bx^2y^{n-2} +by^n, \quad
q(x,y)=ax^n - 2ax^{n-2}y^2 +bxy^{n-1}
\]
obtained in \cite{c1} satisfy a relation of the form $\eta =
\lambda f_1$ where $\lambda=(1/2)\cos 2\theta$. An integrating
factor for this system is given by
\[
\mu(x,y) = ( 1 + 2(ax^{n-1}+by^{n-1}) + (ax^{n-1}-by^{n-1})^2
)^{-(n+3)/(2n-2)}.
\]
We believe that this or similar structure is inherent in most (if
not all) integrating factors and that it can be used to facilitate
the development other integrable forms of \eqref{e2}.

If \eqref{e15} is satisfied then \eqref{e14a} reduces to the
algebraic condition
\begin{equation}
(n+2\alpha+1) \psi + \lambda = 1. \label{e16}
\end{equation}
The second condition is to note that
\begin{equation}
\alpha = \frac{1}{2}(n-3) \label{e17}
\end{equation}
for these integrating factors. With this \eqref{e14c} becomes elementary
and furnishes the relation
\begin{equation}
f_2 = A\eta \label{e18}
\end{equation}
which corresponds to \eqref{e12}. Here $A \ne 0$ is a constant.
Now substituting \eqref{e13}, \eqref{e15} and \eqref{e16} in
\eqref{e14b} we obtain the relation
\[
 \eta = ((n-1)\psi + \lambda)f_1 - A\lambda.
\]
Differentiating this and using \eqref{e15}, \eqref{e16} with
\eqref{e17} gives a differential equation for $f_1$
\[
(1-\lambda)f_1' + \lambda' f_1 = 2A \lambda'.
\]
This can be solved to give
\begin{equation}
f_1 = 2B(\lambda-1) + 2A \label{e19}
\end{equation}
where $B \ne 0$ (the 2 is added for later convenience) is a
constant.

 From the preceding we can see that all functions can be expressed
in terms of $\lambda$ which can be chosen in an arbitrary fashion.
Let
\begin{equation}
N = \big[ \frac{1}{4}(n-1) \big] \label{e20}
\end{equation}
where $[\dots]$ is the greatest integer function and define
\begin{equation}
\lambda(\theta) = a_0 + \sum_{k=1}^N (a_{2k} \cos 2k\theta
+ b_{2k} \sin 2k\theta) \label{e21}
\end{equation}
where $a_0$ and $a_{2k}, b_{2k}$ are arbitrary real numbers such
that $\lambda$ is not constant. In terms of this function we find
that
\begin{equation}
\xi = \frac{B}{n-1}(1-\lambda) \lambda', \quad \eta=B\lambda^2
+ A -B. \label{e22}
\end{equation}
As we noted earlier the degree of $f_1$ given by \eqref{e19} is
only $2N$ rather than the maximal value $n-1$ that it could have
in more general circumstances.

\section{The Abel equation and related integrals}

With $\xi, \eta$ given by \eqref{e22} we can now show that the
trigonometric form \eqref{e4} has constant invariant. Using
$\lambda$ as independent variable, it becomes
\begin{equation}
\frac{d\rho}{d\lambda} = B(\lambda-1)(B\lambda^2 + A - B)\rho^3 +
B(1-3\lambda)\rho^2. \label{e23}
\end{equation}
This equation is quite simple and is rather unusual in that both
relative invariants $s_3, s_5$ defined by \eqref{e7}, \eqref{e8}
are constant. Denoting the coefficients of $\rho^2, \rho^3$ in
\eqref{e23} by $g_2, g_3$ respectively, we find
\[
s_3 = \frac{2}{27}g_2^3 + \frac{1}{3}(g_3g_2' - g_2g_3')
= \frac{2}{27} B^2 (9A-8B)
\]
and
\[
s_5 = (g_2^2 - 3g_3')s_3 = \frac{2}{27}B^3 (4B-3A)(9A-8B).
\]
where the primes denote differentiation with respect to $\lambda$.
From these we have
\begin{equation}
I_1 = \frac{s_5^3}{s_3^5}
= \frac{729}{4} \frac{(4B-3A)^3}{B(9A-8B)^2}. \label{e24}
\end{equation}

Now that we have established that \eqref{e4} is solvable if
\eqref{e22} is satisfied, we proceed to show that these conditions
do produce new center conditions for \eqref{e1}--\eqref{e2}. To do
this we shall obtain the solution of \eqref{e3} using the
integrating factor \eqref{e13}. Since the cartesian form of $\mu$
is not positive on a neighborhood of the origin, we will need to
establish that the solutions themselves are analytic there.

Substituting for $f_1, f_2$ in terms of $\lambda$ and again using
$\lambda$ as independent variable, we have
\begin{align*}
 U(r,\theta)
 &= \int \mu(r,\theta) \xi(\theta) r^n \, d\theta + F(r) \\
 &= \int \frac{B(1-\lambda) r^{2n-2}\, d\lambda}
 {1+ 2(B(\lambda-1)+A)r^{n-1} + A(B \lambda^2 + A -B) r^{2n-2}} + F(r) \\
 &= -B \int \frac{r^{2n-2} x\,dx}
 {ABr^{2n-2}x^2 + 2B(Ar^{n-1}+1)r^{n-1}x + (Ar^{n-1} + 1)^2} + F(r)
\end{align*}
where $x=\lambda-1$ and $F$ is a suitably chosen function of $r$.
The discriminant of the denominator is given by
\[
\Delta = 4B(B-A)(Ar^{n-1}+1)^2 r^{2n-2}.
\]
Within the context of the center--focus problem, we require that a
solution of \eqref{e3} satisfy $1 + r^{n-1}\eta > 0$. Based on
this, we can assume that the sign of $\Delta$ is determined by the
sign of $B(B-A)$. This leads to three separate forms for the
solution $U(r,\theta)=C$.

 We have
\begin{equation}
\begin{aligned}
 U(r,\theta) &= (B-K)\ln | AB \lambda r^{n-1} - KAr^{n-1} + B-K | \\
& \quad - (B+K)\ln | AB \lambda r^{n-1} + KAr^{n-1} + B+K | \quad
\text{if } \Delta>0,
\end{aligned} \label{e25a}
\end{equation}
where $K=\sqrt{B(B-A)}$;
\begin{equation}
\begin{aligned}
 U(r,\theta) &= \ln\Big(AB r^{2n-2} \lambda^2 + 2Br^{n-1} \lambda
 +A(A-B)r^{2n-2} \\
 &\quad + 2(A-B)r^{n-1} + 1\Big) 
 -2K \tan^{-1} (K \frac{Ar^{n-1}\lambda +1}{Ar^{n-1} +1} )
 \quad \text{if } \Delta<0,
\end{aligned}  \label{e25b}
\end{equation}
where $K=\sqrt{B/(A-B)}$; and
\begin{equation}
U(r,\theta) = \ln(Ar^{n-1} \lambda +1) + \frac{Ar^{n-1}+1}{Ar^{n-1}
\lambda +1}  \quad \text{if } \Delta=0. \label{e25c}
\end{equation}
We can easily see that each of these solutions is analytic at
$r=0,$ so its corresponding cartesian form is analytic on a
neighborhood of the critical point $(0,0)$ of \eqref{e1}. This
means that the functions $\xi, \eta$ defined in \eqref{e22} do
produce centers of \eqref{e1}--\eqref{e2} for any odd integer $n
\ge 5$. Also, these solutions generalize to any odd value of $n$
the solutions for the symmetric case for $n=3$ given in \cite{l1}
for the cartesian case and in \cite{c2} for the polar case.

The centers defined for the cases $n=5, 7$ ($N=1$) are symmetric
centers because there always exists a transformation $\theta \to
\theta + \theta_0$ for some constant $\theta_0$ such that
$\lambda$ will become an even function. The functions $\xi, \eta$
in \eqref{e22} will then be respectively odd, even which are
exactly the conditions for the symmetric case. For $n \ge 9$ it is
no longer generally possible to transform $\lambda$ in this
fashion, so the center conditions defined in this way are of a
type not previously recorded.

We note for future reference that the $\Delta=0$ case of \eqref{e25c} is
satisfied by the parameter choice $A=B$. Substituting this in the
invariant \eqref{e24}, we find that $I_1=729/4$. From this it is
easy to see that \eqref{e25a}, \eqref{e25b} give $I_1>729/4$, $I_1<729/4$
respectively.

We summarize the results of the last 2 sections in the following
proposition.

\begin{proposition} \label{prop1}
Let $\xi, \eta$ be given by \eqref{e22} where $\lambda$ is defined
by \eqref{e20}, \eqref{e21}. Then the trigonometric Abel
differential equation \eqref{e4} has constant first absolute
invariant $I_1=s_5^3/s_3^5$ given by \eqref{e24}.
\end{proposition}


\begin{corollary} \label{coro1}
Let $\xi, \eta$ be given by \eqref{e22} where $\lambda$ is defined
by \eqref{e20}, \eqref{e21}. Then the system
\eqref{e1}--\eqref{e2} has a center at the origin for odd integers
$n \ge 5$. For $n=5, 7$ the centers are symmetric centers and for
$n \ge 9$ the centers are new. Each of these center conditions is
integrable and integrals for them are given in polar form by
\eqref{e25a}-\eqref{e25c}.
\end{corollary}


\section{Polynomial solutions}

The usual Poincar\'e solution of \eqref{e2} can be expressed in
the form
 \begin{equation}
U(x,y) = U_2(x,y) + \sum_{k=3}^{\infty} U_k(x,y) = C, \label{e26}
\end{equation}
where $U_2(x,y) = \frac{1}{2}(x^2+y^2)$ and  $U_k(x,y)$ are
homogeneous polynomials of degree $k(n-1)+2$. If the series
terminates for some positive integer $k=M \ge 2,$ the solution is
a polynomial $\mathcal{P}(x,y)$. We can further note that any
integral power $N_1 \ge 2$ of $\mathcal{P}$ will again be a
polynomial solution, but $\mathcal{P}^{N_1}$ will not satisfy a
relation of the basic form \eqref{e26}. Polynomial solutions have
been discussed by other authors \cite{c3,g1}, but they have
restricted their discussion to the case for which $N_1=1$.
However, in the course of this and other related work, we have
found a number of families of solutions whose series solutions
terminate if we take some power $U^{N_1}, N_1 \ge 2$ of the form
\eqref{e26}, but do not if the requirement $N_1=1$ is imposed.
Both constant invariant forms of \eqref{e4} which are discussed in
this paper include polynomial solutions of this type.

Suppose $\xi, \eta$ satisfy \eqref{e5}. Then \eqref{e3} has an
elementary integrating factor of the form $\mu(r)=r^{\alpha}$
where
\begin{equation}
\alpha=-\frac{1}{K}(1+nK). \label{e27}
\end{equation}
The solution is easily given by
\begin{equation}
r^{\alpha+1}\big[1 + \frac{1+\alpha}{n+\alpha} \eta(\theta) r^{n-1}
\big] = C \label{e28}
\end{equation}
where $C$ is a constant. If terms of degree $n+1$ are absent in
$\eta$, the term in the square bracket apart from the 1 is a
homogeneous polynomial of degree $n-1$ in $x, y$. So if $\alpha+1$
is a positive rational number, \eqref{e28} is a polynomial in $x,
y$ or is easily converted to one by taking a suitable power of the
expression. In the case $\alpha=1$ ($K=-1/(n+1)$), $\eta$ can be
chosen to have degree $n+1$ and \eqref{e28} gives the Hamiltonian
solution. An integral of the type \eqref{e28} was given in
\cite{g2}.

The second class of polynomial solutions can be obtained directly
from \eqref{e25a}. The maximal degree of $\lambda$ in the arguments of
the logarithmic terms is $2N$ where $N$ is given by \eqref{e20}.
So we can write $r^{n-1}\lambda = r^{n-1-2N}(r^{2N} \lambda)$.
From this it is obvious that these terms, apart from the constants
($B \pm K$), are homogeneous polynomials of degree $n-1$ in $x,
y$. If the coefficients of the logarithmic terms are negative
rational numbers, $-R_1, -R_2$, the solution can be transformed to
a polynomial. Setting $B-K=-R_1, -(B+K)=-R_2$ where
$K=\sqrt{B(B-A)}$ and solving gives
\begin{equation}
A = \frac{2R_1R_2}{R_1 - R_2}, \quad B = \frac{1}{2}(R_2 - R_1),
 \quad K = \frac{1}{2}(R_1+ R_2). \label{e29}
\end{equation}
If $R_1=P_1/Q_1, R_2=P_2/Q_2$ for positive integers
$P_1,\dots,Q_2$ are given in reduced form, the polynomial solution
is
\begin{equation}
 U(r,\theta) = | AB \lambda r^{n-1}
- KAr^{n-1} + B-K |^{N_1}
 | AB \lambda r^{n-1} + KAr^{n-1} + B+K |^{N_2} \label{e30}
\end{equation}
where $A, B, K$ are given by \eqref{e29}. In this $N_1 = P_1
Q_2/M, N_2 = P_2 Q_1/M$ where $M$ is the greatest
 common divisor of $P_1Q_2, P_2Q_1$.

We now give without proof a cartesian form of the polynomials
given by \eqref{e30}. This second form was derived using a
somewhat different approach (a different integrating factor is
involved), but the two sets are equivalent in that they both can
be shown to produce an integrating factor of the form \eqref{e13}.
We believe that this form is more useful than that given by
\eqref{e30} and it is also valid for $n=3$. All parameters in this
new form are unrelated to those previously used except that $n \ge
3$ is still an odd integer.

Let $m=(n-1)/2$ and define
\[
\beta = \begin{cases}
 \frac{1}{2}m    &\text{if $m$ is even} \\
 \frac{1}{2}(m+1)&\text{if $m$ is odd and } n \ne 3 \\
 0 &\text{if } n=3.
\end{cases}
\]
Let
\[
r(x,y) = \sum_{k=0}^{n-1-2\beta} a_k x^{n-1-2\beta-k} y^k
\]
be a homogeneous polynomial of degree $n-1-2\beta$.
The polynomial solution is given by
\begin{equation}
\mathcal{P}(x,y) = ((x^2+y^2)^{\beta}r(x,y)+1 )^a
 ( K(x^2+y^2)^m - \alpha(x^2+y^2)^{\beta} r(x,y) +1 )^b \label{e31}
\end{equation}
where $\alpha=a/b$ for positive integers $a, b$ and $K \ne 0$ is a
constant. Differentiating \eqref{e31}, we find that
$\mathcal{P}(x,y)$ satisfies \eqref{e2} where
\begin{equation}
\begin{aligned}
 p(x,y) &= \frac{\alpha \beta + m}{m}y(x^2+y^2)^{\beta}r(x,y)
 + \frac{\alpha}{2m}(x^2+y^2)^{\beta+1}\frac{\partial r}{\partial y} \\
 &\quad - \frac{\alpha(\alpha+1)}{2mK}(x^2+y^2)^{2\beta-m}((x^2+y^2)
 \frac{\partial r}{\partial y}
 + 2\beta\, y\, r(x,y) )r(x,y)
\end{aligned} \label{e32}
\end{equation}
 and
\begin{equation}
\begin{aligned}
 q(x,y) &= \frac{\alpha \beta + m}{m}x(x^2+y^2)^{\beta}r(x,y)
 + \frac{\alpha}{2m}(x^2+y^2)^{\beta+1}\frac{\partial r}{\partial x} \\
 &\quad - \frac{\alpha(\alpha+1)}{2mK}(x^2+y^2)^{2\beta-m}((x^2+y^2)
\frac{\partial r}{\partial x}
 + 2\beta\, x\, r(x,y) )r(x,y)
\end{aligned} \label{e33}
\end{equation}
 are homogeneous polynomials of degree $n$. We note that these
forms remain valid for the case $n=3$ ($\beta=0, m=1$).

Polynomials of the types discussed here have appeared previously.
Case(IV) of \cite[Theorem 7]{g1} (when suitably corrected) and Case
(I) of \cite[Theorem 6]{g1} are examples of the constant invariant
case defined by \eqref{e28}. For $n=3$ the polynomial solution
\eqref{e31} is
\begin{equation}
\mathcal{P}(x,y) = (a_0x^2+a_1xy+a_2y^2+1)^a (K(x^2+y^2)
- \alpha(a_0x^2+a_1xy+a_2y^2) +1)^b.
\end{equation}
We believe that the expression given in the Appendix of \cite{c3}
should be of this type.

\section{Discussion}

In this section we briefly discuss the motivation which led to the
results given in this paper. It has  primarily to do with the
values which the first absolute invariant $I_1$ of a constant
invariant Abel equation can take on. If an Abel equation is of
type constant invariant, the solution must have a particular
structure: essentially that given by \eqref{e25a}-\eqref{e25c}.

We start with the constant invariant form given by
\eqref{e5},\eqref{e28}. This leads to a form of \eqref{e4} for
which the value $C$ in \eqref{e9} is given by $ C = -(n-1)K/(Kn -
K - 1)^2$. Solving \eqref{e27} for $K$ and substituting in the
value for $C$, we find that
\[
I_1 - \frac{729}{4} = \frac{19683}{4} \frac{(n-1)^2(\alpha+1)^2
(\alpha+n)^2}  {(2n-1+\alpha)^2(n+1+2\alpha)^2(n-\alpha-2)^2}
\]
which is obviously nonnegative ($\alpha=n-2$ leads to a Bernoulli
equation) for the range of parameter values, $n\ge2, \alpha>-1$,
necessary for a polynomial solution. Moreover, since $\alpha=-1,
-n$ are not allowed values in the solution \eqref{e28} this
expression is always positive. So, within the constraints of the
Poincar\'e problem defined by \eqref{e3}--\eqref{e5}, it is not
possible to obtain solutions of the form \eqref{e25b}, \eqref{e25c}. 
We conclude the discussion of these centers by mentioning (although 
not central to the theme of this paper) that many of the solutions
\eqref{e28} for the Bernoulli form $\alpha=n-2$ are {\it
isochronic}. That is, all trajectories of \eqref{e1} within a
sufficiently small neighborhood of the origin have the same
minimal period.

The second class of constant invariant centers arose by extending
conditions for $n=3$ symmetric centers. We had learned that this
case is solvable in terms of a constant invariant Abel equation
whose invariant can take on any value. This led us to search for
other constant invariant conditions of this type. By using Maple
to solve an extensive system of nonlinear equations, we found
constant invariant conditions for $n=5, 7, 9, 11$. In these, even
though restricted to the symmetric case, the invariant could take
any value, unlike the case of the previous class of centers. It
was the solution of these cases along with the integrating factor
\eqref{e11} and the solutions \eqref{e31} which produced the form
of the integrating factor \eqref{e13}, the conditions \eqref{e15}
and the value of $\alpha$ \eqref{e17}.

The polynomial solutions \eqref{e31} defined by
\eqref{e32},\eqref{e33} lead to an Abel equation whose first
invariant $I_1$ satisfies
\[
I_1 - \frac{729}{4} = \frac{19683}{4}
\frac{\alpha^2(\alpha+1)^2}{(\alpha-1)^2(2\alpha+1)^2(\alpha+2)^2}.
\]
This is similar to the case of the polynomials defined by
\eqref{e5}. Furthermore, these solutions do not generally belong
to the symmetric class of centers for $n \ge 9$. So, because of
this and the discussion in previous paragraph, we felt that there
were likely constant invariant forms of a non--symmetric type for
the odd integers which would produce invariant values $I_1 \le
729/4$.

The results of this paper give sufficient conditions for
\eqref{e4} to have a constant first invariant. We do not know if
there are other constant invariant conditions, but we think it
unlikely. We do believe, however, that it is probable that the
conditions defined by \eqref{e22} are special cases of more
general, integrable center conditions. We hope to consider this
and other applications of conditions like those in \eqref{e15} in
upcoming work.



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