Integrability of vector fields and meromorphic solutions

TL;DR

This paper investigates conditions under which complex foliations with meromorphic solutions admit Liouvillean integrals, showing that many such vector fields on complex manifolds have invariant sets with integrable structures.

Contribution

It establishes that vector fields with meromorphic solutions on complex manifolds induce Liouvillean integrable foliations on invariant surfaces.

Findings

Vector fields with meromorphic solutions have invariant surfaces with Liouvillean first integrals.

On c3b3, rational vector fields with meromorphic solutions admit invariant surfaces with integrable foliations.

The results connect the existence of meromorphic solutions to integrability properties of foliations.

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.