Geodesic
Tensor invariants of natural mechanical systems on compact surfaces, and the corresponding integrals
Sbornik. Mathematics, Tome 188 (1997) no. 2, pp. 307-326
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In this paper sufficient conditions are given for the existence of tensor invariants of geodesic flows to imply their integrability. Analogous assertions are also proved for trajectory automorphisms of geodesic flows. All the tensor invariants on surfaces of constant negative curvature are described. 
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P. I. Topalov. Tensor invariants of natural mechanical systems on compact surfaces, and the corresponding integrals. Sbornik. Mathematics, Tome 188 (1997) no. 2, pp. 307-326. http://geodesic.mathdoc.fr/item/SM_1997_188_2_a6/

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