, Therefore, I U = c U , since I U contains an arc of the leaf c U. Thus, ?? conimplies that the curves ? t are algebraic, as in loc. cit., and their ?-dual curves generate rationally integrable I-angular billiards with a common rational integral, the latter function is constant and thus, I U lies in a leaf

, Some important parts of the work were done during my visits to Sobolev Institute at Novosibirsk and to Tel Aviv University. I wish to thank Andrey Mironov and Misha Bialy for their invitations and hospitality and both institutions for their hospitality and support. I wish to thank Andrey Mironov for his hard work and patience of going through my proofs and helpful remarks. I with to thank Eugenii Shustin, to whom this work is very much due, for helpful discussions, Acknowledgements I am grateful to Misha Bialy and Andrey Mironov for introducing me to polynomially integrable billiards, providing the fundamental first step (their works, vol.10

V. Dragovi´cdragovi´c, E. Ghys, J. Marco, and S. Tabachnikov, I wish to thank Anatoly Fomenko and Elena Kudryavtseva for helpful discussions and for convincing me to extend the results to piecewise smooth case. I wish to thank Sergei Bolotin

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