Integrability and adapted complex structures to smooth vector fields on the plane

Abstract: Singular complex analytic vector fields on the Riemann surfaces enjoy several geometric properties (singular means that poles and essential singularities are admissible). We describe relations between singular complex analytic vector fields $\mathbb{X}$ and smooth vector fields $X$. Our approximation route studies three integrability notions for real smooth vector fields $X$ with singularities on the plane or the sphere. The first notion is related to Cauchy-Riemann equations, we say that a vector field $X$ admits an adapted complex structure $J$ if there exists a singular complex analytic vector field $X$ on the plane provided with this complex structure, such that $X$ is the real part of $\mathbb{X}$. The second integrability notion for $X$ is the existence of a first integral $f$, smooth and having non vanishing differential outside of the singularities of $X$. A third concept is that $X$ admits a global flow box map outside of its singularities, i.e. the vector field $X$ is a lift of the trivial horizontal vector field, under a diffeomorphism. We study the relation between the three notions. Topological obstructions (local and global) to the three integrability notions are described. A construction of singular complex analytic vector fields X using canonical invariant regions is provided.