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 $\mathcal{D}(M)\mathrm{/}{\mathcal{D}}_{\mu }(M)$ 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.