  We deal with the uniqueness of distributional solutions to the continuity
equation with a Sobolev vector field and with the property of being a
Lagrangian solution, that means transported by a flow of the associated
ordinary differential equation. We work in a framework with lack of local
integrability of the solution, in which the classical DiPerna-Lions theory of
uniqueness and Lagrangianity of distributional solutions does not apply due to
the insufficient integrability of the commutator. We introduce a general
principle to prove that a solution is Lagrangian: we rely on a disintegration
along the unique flow and on a new directional Lipschitz extension lemma, used
to construct a large class of test functions in the Lagrangian distributional
formulation of the continuity equation.
