Geodesic flows, bi-Hamiltonian structure and coupled KdV type systems

We show that all the Antonowicz-Fordy type coupled KdV equations have the same symmetry group and similar bi-Hamiltonian structures. It turns out that their configuration space is Diff(S1)⋉C∞(S1)ˆ, where Diff(S1)ˆ is the Bott-Virasoro group of orientation preserving diffeomorphisms of the circle, and all these systems can be interpreted as equations of a geodesic flow with respect to L2 metric on the semidirect product space Diff(S1)⋉C∞(S1)ˆ.