# Commuting vector fields

## TL;DR

This paper investigates the structure of the centralizer of a germ of vector fields at the origin in complex space, with a focus on the special case of two-dimensional vector fields.

## Contribution

It provides new insights into the properties of the centralizer of vector fields, especially in two dimensions, enhancing understanding of their symmetries and invariants.

## Key findings

- Characterization of the centralizer in dimension two
- Identification of special cases with unique properties
- Extension of known results to higher dimensions

## Abstract

In this paper we study the centralizer $C(X)$ of a germ of vector field at $0\in\C^n$. A particular atention is given to the case of dimension two.