ANALYTIC INTEGRABILITY OF A CLASS OF PLANAR POLYNOMIAL DIFFERENTIAL SYSTEMS

In this paper we find necessary and sufficient conditions in order that the differential systems of the form (x) over dot = x f(y), (y) over dot = g(y), with f and g polynomials, have a first integral which is analytic in the variable x and meromorphic in the variable y. We also characterize their analytic first integrals in both variables x and y. These polynomial differential systems are important because after a convenient change of variables they contain all quasi homogeneous polynomial differential systems in R-2.