# New superintegrable models on spaces of constant curvature

## TL;DR

This paper introduces new elementary-function-based angular potentials for 2D superintegrable systems on constant curvature spaces, generalizing TTW and PW models and connecting to Poschl-Teller potentials.

## Contribution

It constructs new two-parameter families of angular potentials explicitly in elementary functions, extending superintegrable models on curved spaces.

## Key findings

- New elementary angular potentials are explicitly constructed.
- The family reduces to Poschl-Teller potential for specific parameters.
- The models generalize TTW and PW systems to curved spaces.

## Abstract

It is known that the fairly (most?) general class of 2D superintegrable systems defined on 2D spaces of constant curvature and separating in (geodesic) polar coordinates is specified by two types of radial potentials (oscillator or (generalized) Kepler ones) and by corresponding families of angular potentials. Unlike the radial potentials the angular ones are given implicitly (up to a function) by, in general, transcendental equation. In the present paper new two-parameter families of angular potentials are constructed in terms of elementary functions. It is shown that for an appropriate choice of parameters the family corresponding to the oscillator/Kepler type radial potential reduces to Poschl-Teller potential. This allows to consider Hamiltonian systems defined by this family as a generalization of Tremblay-Turbiner-Winternitz (TTW) or Post-Winternitz (PW) models both on plane as well as on curved spaces of constant curvature.