Integrability of vector fields and meromorphic solutions

Let $\mathcal F$ be a one-dimensional holomorphic foliation defined on a complex projective manifold and consider a meromorphic vector field $X$ tangent to $\mathcal F$. In this paper, we prove that if the set of integral curves of $X$ that are given by meromorphic maps defined on $\mathbb{C}$ is “large enough”, 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 field whose solutions are meromorphic functions defined on $\mathbb{C}$ admits an invariant analytic set of dimension $2$ such that the restriction of the vector field to it yields a Liouville integrable foliation.