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<p>In this paper we study Riemannian metrics on 2-surfaces with integrable geodesic flows bymeans of an additional rational-in-momenta first integral. This problem is reduced to a quasi-linear system of PDEs. We construct solutions to this system via the classical hodograph method. These solutions give rise to local examples of metrics and rational integrals. Some of the constructed metrics have a very simple form. A family of implicit integrable examples parameterized by two arbitrary functions of one variable is also provided.</p>

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On Rational Integrals of Geodesic Flows on 2-Surfaces

  • 期刊论文

  • 发表日期：

    2025年12月20日

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  • Sergei V. Agapov

摘要

In this paper we study Riemannian metrics on 2-surfaces with integrable geodesic flows bymeans of an additional rational-in-momenta first integral. This problem is reduced to a quasi-linear system of PDEs. We construct solutions to this system via the classical hodograph method. These solutions give rise to local examples of metrics and rational integrals. Some of the constructed metrics have a very simple form. A family of implicit integrable examples parameterized by two arbitrary functions of one variable is also provided.