  On Riemann surfaces $M$, there exists a canonical correspondence between a
possibly multivalued function $\Psi_X$ whose differential is single valued
($i.e.$ an additively automorphic singular complex analytic function) and a
vector field $X$. From the point of view of vector fields, the singularities
that we consider are zeros, poles, isolated essential singularities and
accumulation points of the above. The theory of singularities of the inverse
function $\Psi_X^{-1}$ is extended from meromorphic functions to additively
automorphic singular complex analytic functions. The main contribution is a
complete characterization of when a singularity of $\Psi_X^{-1}$ is either
algebraic, logarithmic or arises from a zero with nonzero residue of $X$.
Relationships between analytical properties of $\Psi_X$, singularities of
$\Psi_X^{-1}$ and singularities of $X$ are presented. Families and sporadic
examples showing the geometrical richness of vector fields on the
neighbourhoods of the singularities of $\Psi_X^{-1}$ are studied. As
applications we have; a description of the maximal univalence regions for
complex trajectory solutions of a vector field $X$, a geometric
characterization of the incomplete real trajectories of a vector field $X$, and
a description of the singularities of the vector field associated to the
Riemann $\xi$ function.
