Nondegenerate singularities of integrable dynamical systems

TL;DR

This paper introduces a new concept of nondegeneracy for singular points in integrable non-Hamiltonian systems, proving local linearizability and rigidity in the analytic case and confirming the conjecture for certain smooth systems.

Contribution

It defines nondegeneracy for singular points in integrable non-Hamiltonian systems and proves local linearizability and deformation rigidity in the analytic case, extending results to smooth systems of specific type.

Findings

Nondegeneracy implies local geometric linearizability.

Nondegenerate singularities are deformation rigid in the analytic case.

The conjecture holds for systems of type (n,0).

Abstract

We give a natural notion of nondegeneracy for singular points of integrable non-Hamiltonian systems, and show that such nondegenerate singularities are locally geometrically linearizable and deformation rigid in the analytic case. We conjecture that the same result also holds in the smooth case, and prove this conjecture for systems of type $(n,0)$, i.e. $n$ commuting smooth vector fields on a $n$-manifold.