  We derive the equations of motion for a planar rigid body of circular shape
moving in a 2D perfect fluid with point vortices using symplectic reduction by
stages. After formulating the theory as a mechanical system on a configuration
space which is the product of a space of embeddings and the special Euclidian
group in two dimensions, we divide out by the particle relabelling symmetry and
then by the residual rotational and translational symmetry. The result of the
first stage reduction is that the system is described by a non-standard
magnetic symplectic form encoding the effects of the fluid, while at the second
stage, a careful analysis of the momentum map shows the existence of two
equivalent Poisson structures for this problem. For the solid-fluid system, we
hence recover the ad hoc Poisson structures calculated by Shashikanth, Marsden,
Burdick and Kelly on the one hand, and Borisov, Mamaev, and Ramodanov on the
other hand. As a side result, we obtain a convenient expression for the
symplectic leaves of the reduced system and we shed further light on the
interplay between curvatures and cocycles in the description of the dynamics.
