Geodesic
On fourth-degree polynomial integrals of the Birkhoff billiard
Informatics and Automation, 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. 
@article{TRSPY_2016_295_a1,
     author = {M. Bialy and A. E. Mironov},
     title = {On fourth-degree polynomial integrals of the {Birkhoff} billiard},
     journal = {Informatics and Automation},
     pages = {34--40},
     publisher = {mathdoc},
     volume = {295},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2016_295_a1/}
}
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M. Bialy; A. E. Mironov. On fourth-degree polynomial integrals of the Birkhoff billiard. Informatics and Automation, Modern problems of mechanics, Tome 295 (2016), pp. 34-40. http://geodesic.mathdoc.fr/item/TRSPY_2016_295_a1/