Moscow Mathematical Journal

Volume 23, Issue 4, October–December 2023 pp. 591–624.

Integrability of Vector Fields and Meromorphic Solutions

Authors: Julio C. Rebelo (1) and Helena Reis (2)
Author institution:(1) Institut de Mathématiques de Toulouse; UMR 5219, Université de Toulouse, 118 Route de Narbonne, F-31062 Toulouse, France
(2) Centro de Matemática da Universidade do Porto, Faculdade de Economia da Universidade do Porto, Portugal

Summary:

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.

2020 Math. Subj. Class. Primary: 34M05, 37F75; Secondary: 34A05.

Keywords: Meromorphic solutions, Liouvillian first integral, foliated Poincaré metric, Riccati and turbulent foliations.

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