Given a pair of smooth divergence free vector fields $X,Y$ on $\mathbb{R}^n$, is there some known characterization whch tells when they do commute?
My approach to solve the problem is to start from the fact that they both preserve the form $\mu=dx^1\wedge ... \wedge dx^n$, then $[X,Y]=0$ if and only if $$0=i_{[X,Y]}\mu = L_Xi_Y\mu =d(i_X i_Y \mu)+i_X(d(i_Y\mu))=d(i_X i_Y \mu).$$
Hence this $(n-2)-$form should be closed and hence I derive some additional condition on them. Do you suggest a different approach to the problem?