First integral by means of the uniformization over the torus

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 $d\psi\circ X = Z\circ\psi$.