the condition that, contrary to (\[eqn:cond\_atoms\]), for some $x \in \mathbb{R}$ the drift measure $\nu$ satisfies $$\nu(\{x\}) > 1/2  \quad (\text{resp., } \nu(\{x\}) < -1/2, \ |\nu(\{x\})| > 1)$$ or $$\nu(\{x\}) = 1/2 \quad (\text{resp., } \nu(\{x\}) = -1/2, \ |\nu(\{x\})| = 1).$$ The basic idea is to provide an insight into the behaviour of the local times $L^X(t,x)$ of solutions $X$ of Eq. (\[eqn:SDE\_mvd\]) in such points $x \in \mathbb{R}$ violating (\[eqn:cond\_atoms\]). This gives rise for an application of Tanaka’s formula, followed by a space transformation, to conclude full information about the features of the solution.\
Throughout the paper, $(\Omega, \mathcal{F}, \mathbf{P})$ stands for a complete probability space endowed with a filtration $\mathbb{F} = (\mathcal{F}_t)_{t \geq 0}$ which satisfies the usual conditions, i.e., $\mathbb{F}$ is right-continuous and $\mathcal{F}_0$ contains all sets from $\mathcal{F}$ which have $\mathbf{P}$-measure zero. For a process $X = (X_t)_{t \geq 0}$ the notation $(X,\mathbb{F})$ indicates that $X$ is $\mathbb{F}$-adapted. The processes considered in the following belong to the class of continuous semimartingales up to a stopping time $S$ and local times of such processes will be an important tool. Given an $\mathbb{F}$-stopping time $S$, we say that $(X,\mathbb{F})$ is a *semimartingale up to $S$* if 