Geodesic
On fourth-degree polynomial integrals of the Birkhoff billiard
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 295 (2016), pp. 34-40.

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We study the Birkhoff billiard in a convex domain with a smooth boundary $\gamma$. We show that if this dynamical system has an integral which is polynomial in velocities of degree $4$ and is independent with the velocity norm, then $\gamma$ is an ellipse. 
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M. Bialy; A. E. Mironov. On fourth-degree polynomial integrals of the Birkhoff billiard. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 295 (2016), pp. 34-40. http://geodesic.mathdoc.fr/item/TM_2016_295_a1/

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