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