Transformation of the Stackel matrices preserving superintegrability

TL;DR

This paper explores how transforming the Stackel matrices in superintegrable systems by adjusting variables' speeds yields an infinite family of new superintegrable systems with explicitly known integrals of motion.

Contribution

It demonstrates that simple variable transformations correspond to trivial changes in the Stackel matrix, generating new superintegrable systems with explicit integrals.

Findings

Infinite family of superintegrable systems derived from transformations

Explicitly defined additional integrals of motion provided

Examples involve angle variables with logarithmic functions

Abstract

If we take a superintegrable Stackel system and make variables "faster" or "slower", that is equivalent to a trivial transformation of the Stackel matrix and potentials, then we obtain an infinite family of superintegrable systems with explicitly defined additional integrals of motion. We present some examples of such transformations associated with angle variables expressed via logarithmic functions.