Nilpotent Systems Admitting an Algebraic Inverse Integrating Factor over C((x,y))

Abstract

We characterize the nilpotent systems whose lowest degree quasihomogeneous term is (y, σ xn)T, σ = ±1, which have an algebraic inverse integrating factor over C((x,y)) . In such cases, we show that the systems admit an inverse integrating factor of the form (h+ . . . )q with h = 2σ xn+1 − (n + 1)y2 and q a rational number. We analyze its uniqueness modulus a multiplicative constant.