Polynomial integrability of natural two-dimensional Hamiltonian systems and applications

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Our project deals with integrable Hamiltonian systems with two degrees of freedom. We study the existence of integrals which are polynomial in momenta. We intend to apply new developed methods in order to construct new phyiscally interesting examples of such systems and tackle a number of famous conjectures. In particular, we plan to solve two conjectures attributed to Birkhoff and the problem of Whittaker. We aim to study the natural generalisations of such systems for magnetic flows and apply obtained results in Finsler geometry and general relativity. Further, we study integrability of magnetic and Finsler billiards.