Geodesic
Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space
Izvestiya. Mathematics , Tome 76 (2012) no. 5, pp. 907-921.

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We consider problems related to the well-known conjecture on the degrees of irreducible polynomial integrals of a reversible Hamiltonian system with two degrees of freedom and toral position space. The main object of study is a special system arising in the analysis of irreducible polynomial integrals of degree 4. In a particular case we have the problem of the motion of two interacting particles on a circle in given potential fields. We prove that if the three potentials are smooth non-constant functions, then this problem has no non-trivial polynomial integrals of arbitrarily high degree. We prove the conjecture completely for systems with a polynomial first integral of degree 4 in the momenta.
Keywords: irreducible integrals, systems with impacts, spectrum of a potential. 
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N. V. Denisova; V. V. Kozlov; D. V. Treschev. Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space. Izvestiya. Mathematics , Tome 76 (2012) no. 5, pp. 907-921. http://geodesic.mathdoc.fr/item/IM2_2012_76_5_a2/

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