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.
