Nondegenerate centers for Abel polynomial differential equations of second kind (Q2357447)

There are given necessary and sufficient conditions for the existence of a center for the system \[ \dot x=(1+e_1x)y+ f_2x^2,\;\dot y=-x- b_2x^2- a_1xy- (c_0+ c_1x) y^2- (d_0+ d_1x)y^3,\tag{1} \] where \(a_i,b_i,c_i,d_i,e_i,f_i\in\mathbb{R}\), when \(f_2=0\) (Theorem 1) and for the case \(a_1=0\) (Theorem 2).