Monodromic Nilpotent Singular Points with Odd Andreev Number and the Center Problem

Given a nilpotent singular point of planar vector field, its monodromy is associated with Andreev number n. The parity n determines whether the existence an inverse integrating factor implies that center. For odd, this not always true. We give characterization for family systems having such center problem cannot be solved by method. Moreover, we study general properties family, determining necessary conditions every and solving in case $$n=3$$ .