On algebraic construction of certain integrable and super-integrable systems

TL;DR

This paper introduces a novel algebraic method for constructing two-dimensional integrable and super-integrable systems, revealing new potentials and integrals of various degrees in momenta, including explicit examples.

Contribution

It presents a new algebraic construction approach for super-integrable systems, identifying three families of potentials with high-degree integrals and providing explicit examples.

Findings

Three families of super-integrable monomial potentials identified

Existence of systems with high-degree integrals in momenta demonstrated

Explicit example of super-integrable system with integrals of degrees 2, 4, and 6

Abstract

We propose a new construction of two-dimensional natural bi-Hamiltonian systems associated with a very simple Lie algebra. The presented construction allows us to distinguish three families of super-integrable monomial potentials for which one additional first integral is quadratic, and the second one can be of arbitrarily high degree with respect to the momenta. Many integrable systems with additional integrals of degree greater than two in momenta are given. Moreover, an example of a super-integrable system with first integrals of degree two, four and six in the momenta is found.