Existence and nonexistence of Puiseux inverse integrating factors in analytic monodromic singularities

In this work, we present some criteria about the existence and nonexistence of both Puiseux inverse integrating factors and Puiseux first integrals for planar analytic vector fields having a monodromic singularity. These functions are a wide generalization of their formal or algebraic counterpart in Cartesian coordinates . We prove that none of the functions and can be used to characterize degenerate centers although the existence of is a sufficient center condition.