# Necessary and sufficient conditions for meromorphic integrability near a curve

## TL;DR

This paper establishes necessary and sufficient conditions for meromorphic integrability of vector fields near algebraic curves, linking Galois theory, monodromy, and normal variational equations to integrability criteria.

## Contribution

It provides a comprehensive characterization of meromorphic integrability near algebraic curves, extending Ayoul-Zung's theorem with new conditions involving monodromy and Galois groups.

## Key findings

- Integrability is characterized by Galois group properties of variational equations.
- Non resonance and Diophantine conditions imply integrability near the curve.
- Results include a linearization theorem for time-dependent vector fields near equilibrium.

## Abstract

Let us consider a vector field $X$ meromorphic on a neighbourhood of an algebraic curve $\bar{\Gamma}\subset \mathbb{P}^n$ such that $\Gamma$ is a particular solution of $X$. The vector field $X$ is $(l,n-l)$ integrable if it there exists $Y_1,\dots,Y_{l-1},X$ vector fields commuting pairwise, and $F_1,\dots,F_{n-l}$ common first integrals. The Ayoul-Zung Theorem gives necessary conditions in terms of Galois groups for meromorphic integrability of $X$ in a neighbourhood of $\Gamma$. Conversely, if these conditions are satisfied, we prove that if the first normal variational equation $NVE_1$ has a virtually diagonal monodromy group $Mon(NVE_1)$ with non resonance and Diophantine properties, $X$ is meromorphically integrable on a finite covering of a neighbourhood of $\Gamma$. We then prove the same relaxing the non resonance condition but adding an additional Galoisian condition, which in fine is implied by the previous non resonance hypothesis. Using the same strategy, we then prove a linearisability result near $0$ for a time dependant vector field $X$ with $X(0)=0\;\forall t$.