On the differentiability of first integrals of two dimensional flows

By using techniques of differential geometry we answer the following open problem proposed by Chavarriga, Giacomini, Giné, and Llibre in 1999. 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 $\gamma_i,\,i=1,2,\cdots,l$, such that $\mathbb R^2\backslash\left(\bigcup^{l}_{i=1}\gamma_i\right)$ has finitely many connected open components, and that on each of these connected sets the system has an analytic first integral. For a homogeneous polynomial differential system in $\mathbb R^3$, there exist finitely many invariant straight lines and invariant conical surfaces such that their complement in $\mathbb R^3$ is the union of finitely many open connected components, and that on each of these connected open components the system has an analytic first integral.