Geodesic flows on diffeomorphism groups with Sobolev metrics and integrable systems

The Harry-Dym equation comes from geodesic flows on diffeomorphism groups. This fact has been observed before by the Marsden school. In this paper we show that the supersymmetric Harry-Dym equation arises from the geodesic flow on the superconformal group. We also show that the stabilizer of a point in the coadjoint representation of the Virasoro algebra endowed with a Sobolev norm consists of a space of projective vector fields. We also show that for each projective vector field, there exists a quadratic that satisfies a Neumann system.