St\"ackel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant Quadrics

TL;DR

This paper explores the relationship between non-degenerate superintegrable systems and associated quadrics, showing how to determine their St"ackel class through these geometric objects, advancing understanding of their classification.

Contribution

It introduces a method to derive the St"ackel class of superintegrable systems from invariant quadrics, linking geometric and algebraic classifications.

Findings

St"ackel class can be obtained from associated quadrics.

Quadrics have position-dependent coefficients reflecting system properties.

Provides a geometric approach to classify superintegrable systems.

Abstract

A non-degenerate second-order maximally conformally superintegrable system in dimension 2 naturally gives rise to a quadric with position dependent coefficients. It is shown how the system's St\"ackel class can be obtained from this associated quadric.The St\"ackel class of a second-order maximally conformally superintegrable system is its equivalence class under St\"ackel transformations, i.e., under coupling-constant metamorphosis.