Families of cubic Thue equations with effective bounds for the solutions

To each non totally real cubic extension $K$ of $\text{Q}$ and to each generator $\alpha$ of the cubic field $K$, we attach a family of cubic Thue equations, indexed by the units of $K$, and we prove that this family of cubic Thue equations has only a finite number of integer solutions, by giving an effective upper bound for these solutions.