Analytic integrability for some degenerate planar vector fields

In this paper we study the analytic integrability of degenerate vector fields of the form (y3+2ax3y+⋯,−x5−3ax2y2+⋯)(y3+2ax3y+⋯,−x5−3ax2y2+⋯) around the origin. For these vector fields it is proved that integrability does not imply formal orbital equivalence to the Hamiltonian leading part. Moreover, it is shown the existence of a system in this class which has a center but is neither analytically integrable nor formal orbital reversible.