# Projectively equivalent 2-dimensional superintegrable systems with projective symmetries

## TL;DR

This paper explores superintegrable systems on 2D geometries sharing the same unparametrized geodesics, introducing a projective equivalence concept and classifying systems with one non-trivial projective symmetry.

## Contribution

It defines projective equivalence for superintegrable systems, analyzes their transformation behavior, and classifies systems with specific projective symmetries.

## Key findings

- Potentials can be reconstructed from a characteristic vector field.
- Potentials of equivalent systems follow a linear superimposition rule.
- Classification of systems with one non-trivial projective symmetry.

## Abstract

This paper combines two classical theories, namely metric projective differential geometry and superintegrability. We study superintegrable systems on 2-dimensional geometries that share the same geodesics, viewed as unparametrized curves. We give a definition of projective equivalence of such systems, which may be considered the projective analog of (conformal) St\"ackel equivalence (coupling constant metamorphosis). Then, we discuss the transformation behavior for projectively equivalent superintegrable systems and find that the potential on a projectively equivalent geometry can be reconstructed from a characteristic vector field. Moreover, potentials of projectively equivalent Hamiltonians follow a linear superimposition rule. The techniques are applied to several examples. In particular, we use them to classify, up to St\"ackel equivalence, the superintegrable systems on geometries with one, non-trivial projective symmetry.