  In this paper, we prove that if $X,Y$ are continuous, Sobolev vector fields
with bounded divergence on the real plane and $[X,Y]=0$, then their flows
commute. In particular, we improve the previous result of Colombo-Tione (2021),
where the authors require the additional assumption of the weak Lie
differentiability on one of the two flows. We also discuss possible extensions
to the $\text{BV}$ setting.
