  We construct divergence-free Sobolev vector fields in
C([0,1];W^{1,r}(T^d;R^d)) with r < d and d >=2 which simultaneously admit any
finite number of distinct positive solutions to the continuity equation. We
then show that the vector fields we produce have at least as many integral
curves starting from a.e. point of T^d as the number of distinct positive
solutions to the continuity equation these vector fields admit. Our work uses
convex integration techniques for the continuity equation introduced in [L.
Szekelyhidi and S. Modena, Annals of PDE, 2018] and [E. Brue, M. Colombo, C. De
Lellis, Archive for Rational Mechanics and Analysis, 2021 ] to study
non-uniqueness for positive solutions. We then infer non-uniqueness for
integral curves from Ambrosio superposition principle.
