Geodesic
Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space
Sbornik. Mathematics, Tome 191 (2000) no. 2, pp. 189-208
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The problem considered here is that of finding conditions ensuring that a reversible Hamiltonian system has integrals polynomial in momenta. The kinetic energy is a zero-curvature Riemannian metric and the potential a smooth function on a two-dimensional torus. It is known that the existence of integrals of degrees 1 and 2 is related to the existence of cyclic coordinates and the separation of variables. The following conjecture is also well known: if there exists an integral of degree $n$ independent of the energy integral, then there exists an additional integral of degree 1 or 2. In the present paper this result is established for $n=3$ (which generalizes a theorem of Byalyi), and for $n=4$, $5$, and $6$ this is proved under some additional assumptions about the spectrum of the potential. 
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N. V. Denisova; V. V. Kozlov. Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space. Sbornik. Mathematics, Tome 191 (2000) no. 2, pp. 189-208. http://geodesic.mathdoc.fr/item/SM_2000_191_2_a1/

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