Complete Resolution Of A Family Of Twisted Thue Equations

One of the first infinite families of Thue equations, $$F_n(X)=X^3-(n-1)X^{2}Y-(n+2)X{Y}^2-Y^3=\pm 1$$ for $n\in \mathbb{Z}$, was solved by Thomas in 1990. This family is associated to the simplest cubic fields $\mathbb{Q}(\lambda )$ of Shanks, where $\lambda$ is a root of $F_n(X,1)$. Levesque and Waldschmidt twisted the Thue equations by an exponent $t$ and looked at the equation $$N_{\mathbb{Q}(\lambda )\mathrm{/}\mathbb{Q}}(X-{\lambda }^{t}Y)=\pm 1,$$ where $t\in \mathbb{Z}$ with $t\mathrm{\ne }0$. In this paper, we find all solutions $(X,Y,n,t)\in {\mathbb{Z}}^4$ with $n,t\in \mathbb{Z}$ and $t\mathrm{\ne }0$ to this family of twisted Thue equations, thereby answering a question of Levesque and Waldschmidt.