Center conditions and integrable forms for the Poincaré problem

We consider the polynomial system d x d t = − y − p ( x , y ) , d y d t = x + q ( x , y ) where p , q are homogeneous polynomials of degree n ⩾ 2 . By finding integrable forms of the system, we obtain several new sets of center conditions valid for general values of n. We also find a general class of centers valid for even values of n which is based on certain parity properties of a related differential equation. We further give several conditions which are valid specifically for n = 4 , 5 . In addition, we demonstrate that the system can be transformed to an Abel equation having rational coefficients and show how the solution to the system leads to the solution of the corresponding Abel equation.