Geodesic
Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities
Izvestiya. Mathematics, Tome 21 (1983) no. 2, pp. 291-306 
V. N. Kolokoltsov. Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities. Izvestiya. Mathematics, Tome 21 (1983) no. 2, pp. 291-306. http://geodesic.mathdoc.fr/item/IM2_1983_21_2_a4/
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     title = {Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities},
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper an explicit description is given for all Riemannian metrics on the sphere and on the torus whose geodesic flows have an additional first integral that is both quadratic in the velocities and independent of the energy integral. Moreover, it is proved that on compact two-dimensional manifolds of higher genus the geodesic flows have no additional polynomial integral. All the results admit straightforward generalizations to arbitrary natural systems given on cotangent bundles of two-dimensional manifolds. Bibliography: 8 titles.

[1] Pidkuiko S. I., Stepin A. M., “Polinomialnye integraly gamiltonovykh sistem”, Dokl. AN SSSR, 239:1 (1978), 50–51 | MR

[2] Darboux G., Lecons sur la théorie générate des surfaces et les applications géométriques du calcul infinitésimal, v. 3, Gauthier–Villars, Paris, 1891; v. 2, 1915

[3] Pars L., Analiticheskaya dinamika, Nauka, M., 1971 | MR

[4] Shiffer M., Spenser D. K., Funktsionaly na konechnykh rimanovykh poverkhnostyakh, IL, M., 1957

[5] Kozlov V. V., “Topologicheskie prepyatstviya k integriruemosti naturalnykh mekhanicheskikh sistem”, Dokl. AN SSSR, 249:6 (1979), 1299–1302 | MR | Zbl

[6] Gurvits A., Kurant R., Teoriya funktsii, Nauka, M., 1968 | MR

[7] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1979 | MR

[8] Tatarinov Ya. V., “Globalnyi vzglyad na dinamiku tverdogo tela. Perekhod k privedennoi sisteme”, Vestn. Mosk. un-ta. Matem., mekhan., 1978, no. 5, 93–98 | MR | Zbl