Polynomial constants of motion for Calogero-type systems in three dimensions

TL;DR

This paper provides explicit formulas for higher-degree polynomial first integrals in three-dimensional Calogero-type systems, demonstrating their maximal superintegrability by combining new and known integrals.

Contribution

It introduces a concise formula for polynomial first integrals in 3D Calogero-type systems, establishing their maximal superintegrability.

Findings

Explicit formulas for polynomial first integrals

Proof of maximal superintegrability

Enhancement of understanding of Calogero-type systems

Abstract

We give an explicit and concise formula for higher-degree polynomial first integrals of a family of Calogero-type Hamiltonian systems in dimension three. These first integrals, together with the already known ones, prove the maximal superintegrability of the systems.