The supersymmetric Camassa–Holm equation and geodesic flow on the superconformal group

We study a family of fermionic extensions of the Camassa-Holm equation.Within this family we identify three interesting classes: (a) equations, whichare inherently hamiltonian, describing geodesic flow with respect to an H^1metric on the group of superconformal transformations in two dimensions, (b)equations which are hamiltonian with respect to a different hamiltonianstructure and (c) supersymmetric flow equations. Classes (a) and (b) have nointersection, but the intersection of classes (a) and (c) gives a candidate fora new supersymmetric integrable system. We demonstrate the Painlev\'e propertyfor some simple but nontrivial reductions of this system.Comment: 14 pages, latex fil