The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor

Abstract We consider analytic families of planar vector fields depending analytically on the parameters in Λ \Lambda that guarantee existence a (may be degenerate and with characteristic directions) monodromic singularity. characterize structure asymptotic Dulac series Poincaré map associated to singularity when family possesses Puiseux inverse integrating factor terms its multiplicity index. This characterization is only valid restricted parameter space \ ∗ \backslash {\Lambda }^{\ast } nonexistence local curves zero angular speed. As byproduct, we are able study center-focus problem (under assumption some Cauchy principal values) very degenerated cases where no other tools available. illustrate theory several nontrivial examples.