  The vortex-wave system is a model for the evolution of 2D incompressible
fluids in which the vorticity is split into a finite sum of Dirac masses plus
an Lp part. Existence of a weak solution for this system was recently proved by
Lopes Filho, Miot and Nussenzveig Lopes, for p > 2, but their result left open
the existence and basic properties of the underlying Lagrangian flow. In this
article we study existence, uniqueness and the qualitative properties of the
(Lagrangian flow for the) linear transport problem associated to the
vortex-wave system. To this end, we study the flow associated to a
two-dimensional vector field which is singular at a moving point. We also
present an approximation scheme for the flow, with explicit error estimates
obtained by adapting results by Crippa and De Lellis for Sobolev vector-fields.
