The talk will be devoted to the problem of decomposing a divergence-free vector measure into a family of measures induced by closed simple curves. We will discuss applications of such decompositions to rigidity properties of vector measures. Moreover, we will demonstrate that in the two-dimensional case such decomposition is possible for any divergence-free vector measure. Ultimately we will discuss some connections between such decompositions and uniqueness of solutions of Cauchy problem for continuity equation. The talk will be based on a recent joint work with P. Bonicatto.