Invariant classification of second-order conformally flat superintegrable systems

TL;DR

This paper classifies all second-order conformally-superintegrable systems in three-dimensional conformally flat spaces, establishing a complete list through algebraic geometry and invariant theory, and describing their conformal classes via a 7-dimensional manifold.

Contribution

It provides a complete classification of nondegenerate superintegrable systems over conformally flat spaces using algebraic and geometric methods, extending previous work.

Findings

The classification confirms the completeness of known systems.

A 7-dimensional manifold parametrizes conformal classes of systems.

The foliation of this manifold is explicitly described using algebraic varieties.

Abstract

In this paper we continue the work of Kalnins et al in classifying all second-order conformally-superintegrable (Laplace-type) systems over conformally flat spaces, using tools from algebraic geometry and classical invariant theory. The results obtained show, through Staeckel equivalence, that the list of known nondegenerate superintegrable systems over three-dimensional conformally flat spaces is complete. In particular, a 7-dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems. This manifold is foliated by the nonlinear action of the conformal group in three-dimensions. Two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation. This foliation is explicitly described using algebraic varieties formed from representations of the conformal group. The proof of these results rely heavily on Groebner basis calculations using the computer algebra software packages Maple and Singular.