Abstract

By using techniques of differential geometry we answer the following open problem: For a given two-dimensional flow what is the maximal order of differentiability of a first integral on a canonical region in function of the order of differentiability of the flow? Moreover, we prove that for every planar polynomial differential system there exist finitely many invariant curves and singular points $\gamma \_i,\; i=1,2,...,l$, such that its complement in $R2$ has finitely many connected open components, and that on each of these connected sets the system has an analytic first integral.