Geodesic
Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry
Sbornik. Mathematics, Tome 189 (1998) no. 10, pp. 1441-1466

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Classical and new results on integrable geodesic flows on two-dimensional surfaces are reviewed. The central question is the classification of such flows up to various equivalences, of which the following four kinds are the most interesting ones: 1) isometry; 2) geodesic equivalence; 3) orbital equivalence; 4) Liouville equivalence. 
@article{SM_1998_189_10_a0,
     author = {A. V. Bolsinov and V. S. Matveev and A. T. Fomenko},
     title = {Two-dimensional {Riemannian} metrics with integrable geodesic flows. {Local} and global geometry},
     journal = {Sbornik. Mathematics},
     pages = {1441--1466},
     publisher = {mathdoc},
     volume = {189},
     number = {10},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_10_a0/}
}
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A. V. Bolsinov; V. S. Matveev; A. T. Fomenko. Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry. Sbornik. Mathematics, Tome 189 (1998) no. 10, pp. 1441-1466. http://geodesic.mathdoc.fr/item/SM_1998_189_10_a0/