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. 
Jonatan Lenells and Gerard Misiolek
math.DG nlin.SI