Varieties of local integrability of analytic differential systems and their applications

In this paper we provide a characterization of local integrability for analytic or formal differential systems in RnRn or CnCn via the integrability varieties. Our result generalizes the classical one of Poincaré and Lyapunov on local integrability of planar analytic differential systems to any finitely dimensional analytic differential systems. As an application of our theory we study the integrability of a family of four-dimensional quadratic Hamiltonian systems.