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<p>We study two-dimensional Riemannian metrics which are superintegrable in the class ofintegrals polynomial in momenta.The study is based on our main technical result, Theorem <InternalRef RefID="FPar3">2</InternalRef>, which states that thePoisson bracket of two integrals polynomial in momenta is an algebraic function ofthe integrals and of the Hamiltonian. We conjecture that two-dimensional superintegrable Riemannian metrics are necessarily real-analytic in isothermal coordinate systems, and give arguments supporting this conjecture. A small modification of the arguments, discussed in the paper, provides a method to construct new superintegrable systems. We prove a special case of the above conjecture which is sufficient to show thatthe metrics constructed by K. Kiyohara [<CitationRef CitationID="CR9">9</CitationRef>], which admit irreducibleintegrals polynomial in momenta, of arbitrary high degree <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation>, are not superintegrable andin particular do not admit nontrivial integrals polynomial in momenta, of degree lessthan <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation>. This result solves Conjectures (b) and (c) explicitly formulated in [<CitationRef CitationID="CR4">4</CitationRef>].</p>

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Real Analyticity of 2-Dimensional Superintegrable Metrics and Solution of Two Bolsinov – Kozlov – Fomenko Conjectures

  • 期刊论文

  • 发表日期：

    2025年8月11日

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  • Vladimir S. Matveev

摘要

We study two-dimensional Riemannian metrics which are superintegrable in the class ofintegrals polynomial in momenta.The study is based on our main technical result, Theorem 2, which states that thePoisson bracket of two integrals polynomial in momenta is an algebraic function ofthe integrals and of the Hamiltonian. We conjecture that two-dimensional superintegrable Riemannian metrics are necessarily real-analytic in isothermal coordinate systems, and give arguments supporting this conjecture. A small modification of the arguments, discussed in the paper, provides a method to construct new superintegrable systems. We prove a special case of the above conjecture which is sufficient to show thatthe metrics constructed by K. Kiyohara [9], which admit irreducibleintegrals polynomial in momenta, of arbitrary high degree \(k\) , are not superintegrable andin particular do not admit nontrivial integrals polynomial in momenta, of degree lessthan \(k\) . This result solves Conjectures (b) and (c) explicitly formulated in [4].