Hidden toric symmetry and structural stability of singularities in integrable systems

TL;DR

This paper develops a systematic approach using hidden torus actions to analyze singularities in integrable systems, demonstrating their stability and classifying associated symmetries, with applications to normal forms and stability of specific singularities.

Contribution

It introduces conditions for the existence and persistence of system-preserving torus actions near singularities, advancing the understanding of their structural stability and symmetry classifications.

Findings

Existence of persistent toric symmetries near singular orbits.

Structural stability of Kalashnikov's parabolic orbits with resonances.

Classification of Hamiltonian torus actions near singularities.

Abstract

The goal of the paper is to develop a systematic approach to the study of (perhaps degenerate) singularities of integrable systems and their structural stability. As the main tool, we use "hidden" system-preserving torus actions near singular orbits. We give sufficient conditions for the existence of such actions and show that they are persistent under integrable perturbations. We find toric symmetries for several infinite series of singularities and prove, as an application, structural stability of Kalashnikov's parabolic orbits with resonances in the real-analytic case. We also classify all Hamiltonian $k$-torus actions near a singular orbit on a symplectic manifold $M^{2n}$ (or on its complexification) and prove that the normal forms of these actions are persistent under small perturbations. As a by-product, we prove an equivariant version of the Vey theorem (1978) about local symplectic normal form of nondegenerate singularities.