Center conditions and integrable forms for the Poincaré problem (Q2019072)

The polynomial system \[ {dx\over dt}=-y-p(x,y),\quad {dy\over dt}=x+q(x,y), \] where \(p\), \(q\) are homogeneous polynomials of degree \(n\geq 2\) is considered. Several sets of center conditions valid for general values of \(n\) are obtained by finding integrable forms of the system. A general class of centers valid for even values of \(n\) which is based on certain parity properties of a related differential equation is also found and several conditions which are valid for \(n=4,5\) are given.