Analytic invariants for the $1:-1$ resonance

TL;DR

This paper introduces two universal analytic invariants for Hamiltonian systems with a non-semisimple 1:-1 resonance, which vanish in integrable cases and are generically non-zero, aiding in the classification of such systems.

Contribution

It constructs two new analytic invariants for Hamiltonian vector fields with 1:-1 resonance, providing tools for their classification and understanding of integrability.

Findings

Two invariants are constructed and shown to be universal.

One invariant vanishes for integrable Hamiltonians.

One invariant is generically non-zero on an open dense set.

Abstract

Associated to analytic Hamiltonian vector fields in $\mathbb{C}^4$ having an equilibrium point satisfying a non semisimple $1:-1$ resonance, we construct two universal constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.