Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \begin{cases} - 1 & \text{ if } X(t,x) >0, \\
1 & \text{ if } X(t,x) < 0 \end{cases}, &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$

This is a typical example of ODE that does not have a solution in the classical sense. The right-hand side is not continuous. However, it is BV, so there exists a regular [Lagrangian flow in the sense of Ambrosio](http://cvgmt.sns.it/media/doc/paper/1612/crippa_bari.pdf). How can it be computed explicitly?