Geodesic
On polynomial integrals of a~mechanical system on a~two-dimensional torus
Izvestiya. Mathematics , Tome 74 (2010) no. 4, pp. 805-817.

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We shall show that if a natural mechanical system defined on a two-dimensional torus and having a real analytic potential possesses a polynomial integral of odd degree in momenta, then the leading coefficients in the momenta satisfy two identities of a special form. We also show that if the system possesses an integral of the fifth degree in momenta, then there exists an integral of the first degree in momenta.
Keywords: integrable Hamiltonian system, polynomial integral. 
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A. E. Mironov. On polynomial integrals of a~mechanical system on a~two-dimensional torus. Izvestiya. Mathematics , Tome 74 (2010) no. 4, pp. 805-817. http://geodesic.mathdoc.fr/item/IM2_2010_74_4_a7/

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