Center conditions for a lopsided quintic polynomial vector field (Q1431475)

The authors determine necessary and sufficient conditions for the polynomial vector field \[ X= -x{\partial\over\partial y}+ y{\partial\over\partial x}+ \sum_{i+j=5} a_{ij} x^i y^j{\partial\over\partial x} \] and its corresponding differential system \[ {dx\over dt}= y+\sum_{i+j=5} a_{ij} x^i y^j,\quad {dy\over dt}=-x, \] to have the origin as a center.