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

Monodromic nilpotent singular points with odd Andreev number and the center problem