In this paper the geodesic flow on a 2-torus in a non-zero magnetic field is considered. Suppose that this flow admits an additional first integral F on N + 2 different energy levels which is polynomial in momenta of arbitrary degree N with analytic periodic coefficients. It is proved that in this case the magnetic field and metrics are functions of one variable and there exists a linear in momenta first integral on all energy levels.
This paper is devoted to searching for Riemannian metrics on 2surfaces whose geodesic flows admit a rational in momenta first integral with a linear numerator and denominator. The explicit examples of metrics and such integrals are constructed. Few superintegrable systems are found having both a polynomial and a rational integrals which are functionally independent of the Hamiltonian.
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