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

First integral by means of the uniformization over the torus