Integrability of invariant metrics on the Virasoro group

Each right-invariant metric on the Virasoro group induces a Hamiltonian vector field on the dual of the Lie algebra virvir equipped with the canonical Lie–Poisson structure. We show that the Hamiltonian vector fields XkXk induced by the metrics given at the identity by the HkHk Sobolev inner products, k⩾0k⩾0, are bi-Hamiltonian relative to a modified Lie–Poisson structure only for k=0k=0 and k=1k=1.