Integrability of vector fields and meromorphic solutions

Abstract

Let $\mathcal{F}$ be a foliation defined on a complex projective manifold $M$ of dimension $n$ and admitting a holomorphic vector field $X$ tangent to it along some non-empty Zariski-open set. In this paper we prove that if $X$ has sufficiently many integral curves that are given by meromorphic functions defined on $\mathbb{C}$ then the restriction of $\mathcal{F}$ to any invariant complex $2$-dimensional analytic set admits a first integral of Liouvillean type. In particular, on ${\mathbb{C}}^3$, every rational vector fields whose solutions are meromorphic functions defined on $\mathbb{C}$ admits a non-empty invariant analytic set of dimension $2$ where the restriction of the vector field yields a Liouvillean integrable foliation.