In this paper, we give a short survey of recent results on the algebraic version of the Birkhoff conjecture for integrable billiards on surfaces of constant curvature. We also discuss integrable magnetic billiards. As a new application of the algebraic technique, we study the existence of polynomial integrals for the two-sided magnetic billiards introduced by Kozlov and Polikarpov.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.
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Bialy et al. (2018) studied this question.
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