Constant invariant solutions of the Poincaré center-focus problem (Q2786022)

The paper deals with the classical Poincaré problem of finding center conditions for the critical point \(O(0,0)\) of the systemNEWLINENEWLINE\[NEWLINE\dot{x}=-y-p(x,y), \qquad \dot{y}=x+q(x,y),NEWLINE\]NEWLINENEWLINEwhere \(p\), \(q\) are homogeneous polynomials of degree \(n\geq2\). The underlying idea is the previously obtained knowledge of certain conditions which produce a constant first absolute invariant of an Abel equation in polar coordinates associated with that system.