Analytic integrability and characterization of centers for nilpotent singular points

A method which provides necessary conditions to obtain a local analytic first integral in a neighborhood of a nilpotent singular point is developed. As an application we provide sufficient conditions in order that systems of the form where P_n and Q_n are homogeneous polynomials of degree n = 2, 3, 4, 5 have a local analytic first integral of the form H=y²+F(x, y), where F starts with terms of order higher than 2. We remark that, in general, the existence of such integral is only guaranteed when the singular point is a nilpotent center and the system has a formal first integral, see 6]. Therefore, we characterize the nilpotent centers of systems which have a local analytic first integral.