Integrability of Centers Perturbed by Quasi-Homogeneous Polynomials

The objective of this work is to give sufficient and necessary conditions for integrability of differential equation systems in the plane of the form ˙xsyy2 py1 qX2 x, y., ˙ysx2 qy1qY2 x, y., where X2 x, y. and Y2 x, y. are pyq. quasihomogeneous polynomials of degree 2 pqyqq1 and 2 pqypq1, respectively.