Non-formally integrable centers admitting an algebraic inverse integrating factor

We study the existence of a class of inverse integrating factor for a family of non-formally integrable systems whose lowest-degree quasi-homogeneous term is a Hamiltonian vector field. Once the existence of an inverse integrating factor is established, we study the systems having a center. Among others, we characterize the centers of the perturbations of the system \begin{document}$ -y^3\partial_x+x^3\partial_y$\end{document} having an algebraic inverse integrating factor.