On First Integrals for Holomorphic Vector Fields

TL;DR

This paper provides a geometric proof for the existence of holomorphic first integrals in specific vector fields within complex spaces, advancing understanding of their integrability properties.

Contribution

It offers a new geometric demonstration of holomorphic first integrals for certain vector fields in complex dimensions two and three.

Findings

Existence of holomorphic first integrals proven geometrically for specific vector fields.

Method applicable to vector fields in ^2 and ^3.

Enhances understanding of integrability in complex dynamical systems.

Abstract

We give a geometrical demonstration to the existence of holomorphic first integrals for certain kind of vector fields in $\mathbb{C}^2$ and $\mathbb{C}^3$.