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

We study two-dimensional Riemannian metrics which are superintegrable in the class of integrals polynomial in momenta. The study is based on our main technical result, Theorem 2, which states that the Poisson bracket of two integrals polynomial in momenta is an algebraic function of the 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 that the metrics constructed by K. Kiyohara [9], which admit irreducible integrals polynomial in momenta, of arbitrary high degree $$k$$ , are not superintegrable and in particular do not admit nontrivial integrals polynomial in momenta, of degree less than $$k$$ . This result solves Conjectures (b) and (c) explicitly formulated in [4].