Geodesic
Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 74 (2019) no. 2, pp. 187-209

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Magnetic billiards in a convex domain with smooth boundary on a constant-curvature surface in a constant magnetic field is considered in this paper. The question of the existence of an integral of motion which is a polynomial in the components of the velocity is investigated. It is shown that if such an integral exists, then the boundary of the domain defines a non-singular algebraic curve in $\mathbb{C}^3$. It is also shown that for a domain other than a geodesic disk, magnetic billiards does not admit a polynomial integral for all but perhaps finitely many values of the magnitude of the magnetic field. To prove our main theorems a new dynamical system, ‘outer magnetic billiards’, on a constant-curvature surface is introduced, a system ‘dual’ to magnetic billiards. By passing to this dynamical system one can apply methods of algebraic geometry to magnetic billiards. Bibliography: 30 titles.
Keywords: magnetic billiards, constant-curvature surfaces, polynomial integrals. 
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     title = {Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane},
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M. Bialy; A. E. Mironov. Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 74 (2019) no. 2, pp. 187-209. http://geodesic.mathdoc.fr/item/RM_2019_74_2_a0/