Existence of an inverse integrating factor, center problem and integrability of a class of nilpotent systems

We characterize the nilpotent systems whose lowest degree quasi-homogeneous term is (y, σxn)T, σ = ±1, having a formal inverse integrating factor. We prove that, for n   even, the systems with formal inverse integrating factor are formally orbital equivalent to (x˙,y˙)T=(y,xn)T. In the case n odd, we give a formal normal form that characterizes them. As a consequence, we give the link among the existence of formal inverse integrating factor, center problem and integrability of the considered systems.