  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.
