Formal Weierstrass Nonintegrability Criterion for Some Classes of Polynomial Differential Systems in C²

In this paper, we present a criterion for determining the formal Weierstrass nonintegrability of some polynomial differential systems in the plane C². The criterion uses solutions of the form y = f(x) of the differential system in the plane and their associated cofactors, where f(x) is a formal power series. In particular, the criterion provides the necessary conditions in order that some polynomial differential systems in C² would be formal Weierstrass integrable. Inside this class there exist non-Liouvillian integrable systems. Finally we extend the theory of formal Weierstrass integrability to Puiseux Weierstrass integrability.