On a family of unit equations over simplest cubic fields

Let $a\in \mathbb{Z}$ and $\rho$ be a root of $f_a(x)=x^3-ax^2-(a+3)x-1$, then the number field $K_a=\mathbb{Q}(\rho)$ is called a simplest cubic field. In this paper we consider the family of unit equations $u_1+u_2=n$ where $u_1,u_2\in \mathbb{Z}[\rho]^*$ and $n\in \mathbb{Z}$. We completely solve the unit equations under the restriction $|n|\leq \max\{1,|a|^{1/3}\}$.