Let $U$ be some open set in $\mathbb{R}^N$, and suppose $X_1,\ldots, X_K$ are a collection of smooth vector fields mapping $U \to \mathbb{R}^N$.
Is there a simple condition which characterizes when there exists, at least locally, a nowhere-zero vector field $Y$ such that the Lie brackets $[X_k, Y] = 0$, for all $k =1 ,\ldots, K$?
If such a $Y$ exists then $Y$ commutes with each $[X_j, X_k]$ but this seems to not be obviously useful. This smells like it's somehow just the Frobenius theorem but I'm not familiar enough to be able to make the connection as clear as I'd like.