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, which are inherently Hamiltonian, describing geodesic flow with respect to an H¹ metric on the group of superconformal transformations in two dimensions, (b) equations which are Hamiltonian with respect to a different Hamiltonian structure and (c) supersymmetric equations. Classes (a) and (b) have no intersection, but the intersection of classes (a) and (c) gives a system with interesting integrability properties. We demonstrate the Painlevé property for some simple but nontrivial reductions of this system, and also discuss peakon-type solutions.