+ g(x-)\right)/2), \qquad x \in \mathbb{R},$$ admits a unique càdlàg solution. This solution $g$ is strictly positive and the strictly increasing and continuous primitive $G(x) = \int_0^x g(y)\, {\mbox{\upshape d}}y$ transforms Eq. (\[eqn:SDE\_mvd\]) into an equation without drift (cf. [@engelbert_schmidt:1985], Proposition 1, or [@engelbert_schmidt:1989_III], Proposition (4.29)). In the several cases, the explicit form of the solution to (\[eqn:integral\_eqn\]) can be found in [@engelbert_schmidt:1989_III], (4.26), (4.26$^\prime$) and (4.26$^{\prime\prime}$), respectively.\
The case $\nu(\{x\}) = 1/2$ (resp., $\nu(\{x\}) = -1/2$, $|\nu(\{x\})| = 1$) for some $x \in \mathbb{R}$ is excluded in (\[eqn:integral\_eqn\]) since it corresponds, as we will see, to a reflecting barrier at the point $x$, which requires different methods to treat Eq. (\[eqn:SDE\_mvd\]) than by assuming (\[eqn:cond\_atoms\]) (cf. W. Schmidt [@schmidt:1989], R.F. Bass and Z.-Q. Chen [@bass_chen]). Moreover, in the case that $\nu(\{x\}) > 1/2$ (resp., $\nu(\{x\}) < -1/2$, $|\nu(\{x\})| > 1$) holds for some $x \in \mathbb{R}$, in general, there is no solution to Eq. (\[eqn:SDE\_mvd\]).\
Indeed, in their famous paper on skew Brownian motion, J.M. Harrison and L.A. Shepp [@harrison_shepp] studied Eq. (\[eqn:SDE\_mvd\]) with symmetric local time in the special case $X_0 = 0$, $b \equiv 1$ and $\nu = \beta \, \delta_0$, where $\delta_0$ is the Dirac measure 