The given SDE is a special case of a one-dimensional SDE with a **drift measure**
$$
dX = \int_{\mathbb{R}} d \Lambda_X(t,x) \mu(dx) + \sigma(X(t)) dZ(t) \tag{$\star$}
$$
where from left: $\Lambda_Y(t,x)$ is the (symmetric) local time of $X(t)$ at the level $x$, $\mu$ is a measure that we will specify shortly, $\sigma$ is a measurable positive function, and $Z$ is the OP's notation for Brownian motion. For example, ($\star$) reduces to a standard one-dimensional SDE (without local time terms) when the measure $\mu$ is absolutely continuous with respect to Lebesgue measure and $d \mu / d \lambda = b/\sigma^2$ where $\lambda$ is Lebesgue measure on $\mathbb{R}$.

For simplicity, assume that the singular continuous part of the measure $\mu$ is zero and write
$$
\mu(dx) = \phi(x) \lambda(dx) + \sum_{i} (2 a_i - 1) \delta_{x_i} (dx)
$$
where $\phi$ is assumed to be measurable. Basically, we decomposed $\mu$ into a part that is absolutely continuous with respect to Lebesgue measure and another part which is discrete, i.e., a (countable) sum of point masses. Substituting this decomposition back into ($\star$) gives the SDE:
$$
dX = \phi \sigma^2 dt + \sum_{i} (2 a_i - 1) d \Lambda_X(t,x_i) + \sigma(X(t)) dZ(t) \tag{$\star \star$}
$$
Please note that ($\star \star$) is allowed to have countable number of local time terms. In this general context, strong existence and uniqueness was proven by J.- F. Le Gall (1984); see below for a detailed citation. I included a few more related works, which might be useful. To be sure, the SDE given by the OP is a special case of ($\star \star$) with a single local time term at $x_0 = a$ and $a_0 = 1$.

**References**

*Strong Existence and Uniqueness Result*

Le Gall, J.-F. **One-dimensional stochastic differential equations involving the local times of the unknown process.** Stochastic analysis and applications (Swansea, 1983), 51–82, Lecture Notes in Math., 1095, Springer, Berlin, 1984.

*Closely Related Works*

Bass, Richard F., and Zhen-Qing Chen. **One-dimensional stochastic differential equations with singular and degenerate coefficients.** Sankhyā: The Indian Journal of Statistics (2005): 19-45.
APA

Lejay, Antoine, and Miguel Martinez. **A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients.** The Annals of Applied Probability (2006): 107-139.

Étoré, Pierre. **On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients.** Electron. J. Probab 11.9 (2006): 249-275.