  We prove a Lipschitz extension lemma in which the extension procedure
simultaneously preserves the Lipschitz continuity for two non-equivalent
distances. The two distances under consideration are the Euclidean distance
and, roughly speaking, the geodesic distance along integral curves of a
(possibly multi-valued) flow of a continuous vector field. The Lipschitz
constant for the geodesic distance of the extension can be estimated in terms
of the Lipschitz constant for the geodesic distance of the original function.
This Lipschitz extension lemma allows us to remove the high integrability
assumption on the solution needed for the uniqueness within the DiPerna-Lions
theory of continuity equations in the case of vector fields in the Sobolev
space $W^{1,p}$, where $p$ is larger than the space dimension, under the
assumption that the so-called "forward-backward integral curves" associated to
the vector field are trivial for almost every starting point. More precisely,
for such vector fields we prove uniqueness and Lagrangianity for weak solutions
of the continuity equation that are just locally integrable.
