in zero, and they proved that there is no solution for the case $|\beta|>1$.\
Referring to [@harrison_shepp], J.F. Le Gall [@LeGall_1984] (see after the proof of [@LeGall_1984], Theorem 2.3) asserted, without giving a proof, that in his context ($b$ is of finite variation and bounded from below by a strictly positive constant) the result on the non-existence of a solution started at $x_0$ can be extended to Eq. (\[eqn:SDE\_mvd\]) with symmetric local time if $|\nu(\{x_0\})| > 1$.\
Also for symmetric local time, R.F. Bass and Z.-Q. Chen [@bass_chen] stated some propositions if $|\nu(\{x\})| > 1$ or $|\nu(\{x\})| = 1$ for some $x \in \mathbb{R}$ under the assumption, besides others, that the diffusion coefficient $b$ is bounded below by a positive constant (see [@bass_chen], Theorem 3.2 and 3.3). However, in both formulations and proofs, we feel that there is not enough clarity which had enabled us to follow their arguments. In particular, they have not pointed out where their assumptions on $b$ are used. But we shall see below that for general diffusion coefficient $b$ their Theorem 3.2 does not remain true.\
The purpose of the present note is to give a general, complete and rigorous approach to Eq. (\[eqn:SDE\_mvd\]) under 