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JOURNALS // Sibirskii Matematicheskii Zhurnal // Archive

Sibirsk. Mat. Zh., 2023 Volume 64, Number 5, Pages 881–894 (Mi smj7803)

On some properties of semi-Hamiltonian systems arising in the problem of integrable geodesic flows on the two-dimensional torus

S. V. Agapovab, Zh. Sh. Fakhriddinova

a Novosibirsk State University
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: Bialy and Mironov demonstrated in a recent series of works that the search for polynomial first integrals of a geodesic flow on the 2-torus reduces to the search for solutions to a system of quasilinear equations which is semi-Hamiltonian. We study the various properties of this system.

Keywords: integrable geodesic flow, polynomial first integral, weakly nonlinear system, semi-Hamiltonian system, Riemann invariants, generalized godograph method, Euler–Poisson–Darboux equation.

UDC: 517.938

MSC: 35R30

Received: 14.04.2023
Revised: 02.05.2023
Accepted: 16.05.2023

DOI: 10.33048/smzh.2023.64.501



© Steklov Math. Inst. of RAS, 2025