# First integral by means of the uniformization over the torus

## TL;DR

This paper generalizes the uniformization method over the torus to establish the existence of first integrals for plane vector fields, extending Finn's 1973 approach to classify differential operators.

## Contribution

It introduces a generalized uniformization technique over the torus for analyzing the existence of first integrals in plane vector fields.

## Key findings

- Unified approach to uniformization over the torus
- Application to existence of first integrals
- Extension of Finn's classification method

## Abstract

The uniformization of a direction field was defined by Finn (in 1973) for the classification of certain differential operators. In the present note we recover the idea of uniformization in order to generalize it and apply it to the existence of first integrals. Roughly speaking, we say that a plane vector field $X$ on $D\subset\mathbb{R}^2$ is uniformizable over $T^2$ (the torus) if there is a constant vector field $Z$ on $\Delta\subset T^2$ and a diffeomorphism $\psi\colon D\to \Delta\subset T^2$ such that $d\psi\circ X = Z\circ\psi$.