  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.
