On the Drach superintegrable systems

TL;DR

The paper analyzes cubic invariants in two-dimensional degenerate Hamiltonian systems through separation variables tied to Stäckel problems with quadratic integrals. It shows that, for superintegrable Stäckel systems, the cubic invariant admits a compact algebro-geometric representation and provides a complete catalog of planar systems known to possess a cubic invariant. By recasting the cubic invariant as a generalized angular momentum built from separation variables, the authors derive a nonlinear oscillator-like algebra and connect eight Drach Hamiltonians to the Stäckel family, detailing their cases and Lax representations. The work extends the framework to additional degenerate plane systems in multiple coordinate charts and highlights potential generalizations to higher dimensions, with implications for the classification of superintegrable systems.

Abstract

Cubic invariants for two-dimensional degenerate Hamiltonian systems are considered by using variables of separation of the associated Stäckel problems with quadratic integrals of motion. For the superintegrable Stäckel systems the cubic invariant is shown to admit new algebro-geometric representation that is far more elementary than the all the known representations in physical variables. A complete list of all known systems on the plane which admit a cubic invariant is discussed.