Local high-degree polynomial integrals of geodesic flows and the generalized hodograph method

Abstract

We study Riemannian metrics on 2-surfaces with integrable geodesic flows such that an additional first integral is high-degree polynomial in momenta. This problem reduces to searching for solutions to certain quasi-linear systems of PDEs which turn out to be semi-Hamiltonian. We construct plenty of local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5. Our construction is essentially based on applying the generalized hodograph method.

References in corpus (3)

• Invariant characterization of Liouville metrics and polynomial integrals
• Hydrodynamic-type systems describing 2-dimensional polynomially integrable geodesic flows
• Integrable magnetic geodesic flows on 2-surfaces