Local analytic integrability for nilpotent centers

Let X(x, y) and Y(x, y) be real analytic functions without constant and linear terms defined in a neighborhood of the origin. Assume that the analytic differential system x = y + X(x,y), y = Y(x, y) has a nilpotent center at the origin. The first integrals, formal or analytic, will be real except if we say explicitly the converse. We prove the following. (a) If X = yf (x, y2) and Y = g (x, y2), then the system has a local analytic first integral of the form H = y2+ F(x, y), where F starts with terms of order higher than two. (b) If the system has a formal first integral, then it has a formal first integral of the form H = y2+ F(x, y), where F starts with terms of order higher than two. In particular, if the system has a local analytic first integral defined at the origin, then it has a local analytic first integral of the form H = y2+ F(x, y), where F starts with terms of order higher than two. (c) As an application we characterize the nilpotent centers for the differential systems ẋ = y + P3(x, y), ẏ = Q3(x, y), which have a local analytic first integral, where P3and Q3are homogeneous polynomials of degree three.