  Helhmoltz-Kirchhoff equations of motions of vortices of an incompressible
fluid in the plane define a dynamics with singularities and this leads to a
Zermelo navigation problem describing the ship travel in such a field where the
control is the heading angle. Considering one vortex, we define a time
minimization problem which can be analyzed with the technics of geometric
optimal control combined with numerical simulations, the geometric frame being
the extension of Randers metrics in the punctured plane, with rotational
symmetry. Candidates as minimizers are parameterized thanks to the Pontryagin
Maximum Principle as extremal solutions of a Hamiltonian vector field. We
analyze the time minimal solution to transfer the ship between two points where
during the transfer the ship can be either in a strong current region in the
vicinity of the vortex or in a weak current region. The analysis is based on a
micro-local classification of the extremals using mainly the integrability
properties of the dynamics due to the rotational symmetry. The discussion is
complex and related to the existence of an isolated extremal (Reeb) circle due
to the vortex singularity. The explicit computation of cut points where the
extremal curves cease to be optimal is given and the spheres are described in
the case where at the initial point the current is weak.
