# Local normal forms of dynamical systems with a singular underlying geometric structure

## TL;DR

This paper establishes conditions under which local normal forms can be simultaneously achieved for vector fields and singular geometric structures, extending classical normalization results to broader singular contexts.

## Contribution

It introduces a unified approach for local normalization of vector fields coupled with various singular geometric structures, including volume, symplectic, Poisson, and contact forms, using toric methods.

## Key findings

- Existence of formal simultaneous normalization for singular structures and vector fields.
- Analytic normalization when structures and vector fields are analytic and integrable.
- Application of toric and equivariant methods to normalization problems.

## Abstract

In this paper we prove the existence of a simultaneous local normalization for couples $(X,\mathcal{G})$, where $X$ is a vector field which vanishes at a point and $\mathcal{G}$ is a singular underlying geometric structure which is invariant with respect to $X$, in many different cases: singular volume forms, singular symplectic and Poisson structures, and singular contact structures. Similarly to Birkhoff normalization for Hamiltonian vector fields, our normalization is also only formal, in general. However, when $\mathcal{G}$ and $X$ are (real or complex) analytic and $X$ is analytically integrable or Darboux-integrable then our simultaneous normalization is also analytic. Our proofs are based on the toric approach to normalization of dynamical systems, the toric conservation law, and the equivariant path method. We also consider the case when $\mathcal{G}$ is singular but $X$ does not vanish at the origin.