Geometric interpretation of the vanishing Lie Bracket for two-dimensional rough vector fields

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 BV setting.