Center conditions of a particular polynomial differential system with a nilpotent singularity (Q2287236)

It is well-known that not all nilpotent centers of planar analytic vector fields are locally analytic integrable but all them are orbitally equivalent to time-reversible ones. Based on this fact, the authors develop an orbital reversibility algorithm to obtain the orbital reversible obstructions to have a center for any planar vector field with a nilpotent singularity. As an application they study the analytic integrability and the center problem at the origin of the family of real planar vector fields \[ \dot x= y +a xy +b x^{2q+2},\quad \dot y=-x^{4q+1}+ cy^2+d x^{2q+1}y+ex^{4q+2},\] where \(q\) is a natural number and \(a,b,c,d\) and \(e\) are real parameters. Observe that this differential equation is defined by a vector field which is the sum of two quasi-homogeneous vector fields. This family was introduced several years ago by J. Torregrosa and the reviewer as one of the simplest families with a nilpotent singularity where the center problem is open. In the reviewed paper the authors give a list of centers for any value of \(q,\) conjecture that they are the only ones, and prove their own conjecture for \(q\le100.\)