Analytic invariant curves of planar analytic vector fields

In this paper, we analyze the existence and uniqueness of the analytic invariant curves passing through any singular point of any planar vector field based on its leading quasi-homogeneous term. As a consequence, under generic conditions of non-resonance, we characterize all irreducible analytic invariant curves at the origin of any analytic vector field. For differential systems satisfying such generic conditions, we solve the local analytic integrability problem and the monodromy problem for polynomial families of autonomous differential systems.