Geodesic
Amari--Chentsov connections and their geodesics on homogeneous spaces of diffeomorphism groups
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 49-62

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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. 
@article{ZNSL_2013_411_a2,
     author = {J. Lenells and G. Misio{\l}ek},
     title = {Amari--Chentsov connections and their geodesics on homogeneous spaces of diffeomorphism groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {49--62},
     publisher = {mathdoc},
     volume = {411},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a2/}
}
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J. Lenells; G. Misiołek. Amari--Chentsov connections and their geodesics on homogeneous spaces of diffeomorphism groups. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 49-62. http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a2/