Amari–Chentsov connections and their geodesics on homogeneous spaces of diffeomorphism groups

We study the family of $\alpha$-connections of Amari–Chentsov on the homogeneous space $\mathcal D(M)/\mathcal D_\mu(M)$ of diffeomorphisms modulo volume-preserving diffeomorphims of a compact manifold $M$. We show that in some cases their geodesic equations yield completely integrable Hamiltonian systems.