Geodesic
Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus
Matematičeskie zametki, Tome 64 (1998) no. 1, pp. 37-44.

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We consider dynamical systems with two degrees of freedom whose configuration space is a torus and which admit first integrals polynomial in velocity. We obtain constructive criteria for the existence of conditional linear and quadratic integrals on the two-dimensional torus. Moreover, we show that under some additional conditions the degree of an “irreducible” integral of the geodesic flow on the torus does not exceed 2. 
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N. V. Denisova. Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus. Matematičeskie zametki, Tome 64 (1998) no. 1, pp. 37-44. http://geodesic.mathdoc.fr/item/MZM_1998_64_1_a4/

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