Integrability of a linear center perturbed by a fifth degree homogeneous polynomial (Q1385180)

The authors study the integrability of planar polynomial differential systems: \[ \dot x= -y+ X_5(t,y),\quad \dot y= x+ Y_5(x, y),\tag{1} \] where \(X_5\) and \(Y_5\) are homogeneous polynomials of \(x\) and \(y\) with constant coefficients. They obtain four sets of complicated conditions for (1) to be integrable. They give a conjecture for the vanishing of all Lyapunov constants, i.e., the angular parameters and the radial ones must be independent. If this conjecture is true, then the integrable cases found by them will be the only possible ones.