Real-analyticity of 2-dimensional superintegrable metrics and solution of two Bolsinov-Kozlov-Fomenko conjectures

TL;DR

This paper investigates the real-analytic nature of 2D superintegrable metrics, proves a special case of a conjecture, and solves two longstanding conjectures related to polynomial integrals in superintegrable systems.

Contribution

It proves a special case of the conjecture that superintegrable metrics are real-analytic and resolves two conjectures from Bolsinov, Kozlov, and Fomenko (1995).

Findings

Proves a special case of the real-analyticity conjecture.

Shows metrics by Kiyohara are not superintegrable.

Provides methods to construct new superintegrable systems.

Abstract

We study two-dimensional Riemannian metrics which are superintegrable in the class of polynomial in momenta integrals. The study is based on our main technical result, Theorem 3, which states that the Poisson bracket of two polynomial in momenta integrals is an algebraic function of the integrals and of the Hamiltonian. We conjecture that two-dimensional superintegrable Riemannian metrics are necessary real-analytic in isothermal coordinate systems, and give arguments supporting this conjecture. Small modification of the arguments, discussed in the paper, provides a methods 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 in 2001, which admit irreducible polynomial in momenta integrals of arbitrary high degree $k$, are not superintegrable and in particular do not admit nontrivial polynomial in momenta integral of degree less than $k$. This result solves Conjectures (b) and (c) explicitly formulated in Bolsinov, KOzlov and Fomenko in 1995.