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MR675528 (84d:58064) 58F17 (53C22 53C45 58F05 58G30)
Kolokolʹtsov, V. N. Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial with respect to velocities. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 994--1010, 1135.
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