  We consider the continuity equation $\partial_t \mu_t +
\mathop{\mathrm{div}}(b \mu_t) = 0$, where $\{\mu_t\}_{t \in \mathbb R}$ is a
measurable family of (possibily signed) Borel measures on $\mathbb R^d$ and $b
\colon \mathbb R \times \mathbb R^d \to \mathbb R^d$ is a bounded Borel vector
field (and the equation is understood in the sense of distributions). If the
measure-valued solution $\mu_t$ is non-negative, then the following
\emph{superposition principle} holds: $\mu_t$ can be decomposed into a
superposition of measures concentrated along the integral curves of $b$. For
smooth $b$ this result follows from the method of characteristics, and in the
general case it was established by L. Ambrosio. A partial extension of this
result for signed measure-valued solutions $\mu_t$ was obtained in \cite{AB},
where the following problem was proposed: does the superposition principle hold
for signed measure-valued solutions in presence of unique flow of
homeomorphisms solving the associated ordinary differential equation? We answer
to this question in the negative, presenting two counterexamples in which
uniqueness of the flow of the vector field holds but one can construct
non-trivial signed measure-valued solutions to the continuity equation with
zero initial data.
