While researching on PDE's I came to the following question:
To a given differentiable vector field $X_1$ on an open, simply connected subset $\Omega$ of $\mathbb R^n$, is it possible to find linear independent vector fields $X_2,\dots,X_n$ (possibly vanishing at the origin), such that $[X_1,X_i]=0$ for all $i=1,\cdots,n$ where $[\cdot,\cdot]$ is the Lie bracket.
The vector field $X_1$ is inducing an exponentially stable flow on $\Omega$ with the origin as fix point. I was wondering if there exists some sufficient conditions for the existence of such vector fields.
I was thinking, that if $\Phi^t$ (the flow related to $X_1$) is contractive for all $t$ and $n $ is even, one can take a ball in $\Omega$ around the origin and use a push forward under $\Phi^t$ of its tangent space. However, this is too resctrictive.