INTEGRABILITY OF VECTOR FIELDS AND MEROMORPHIC SOLUTIONS

Let F be a one-dimensional holomorphic foliation defined on a complex projective manifold and consider a meromorphic vector field X tangent to F. In this paper, we prove that if the set of integral curves of X that are given by meromorphic maps defined on C is "large enough", then the restriction of F to any invariant complex 2-dimensional analytic set admits a first integral of Liouvillean type. In particular, on C-3, every rational vector field whose solutions are meromorphic functions defined on C admits an invariant analytic set of dimension 2 such that the restriction of the vector field to it yields a Liouville integrable foliation.