Elementary resolution of a family of quartic Thue equations over function fields

We consider and completely solve the parametrized family of Thue equations \begin{eqnarray*}X(X-Y)(X+Y)(X-\lambda Y)+Y^4=\xi,\end{eqnarray*} where the solutions $x,y$ come from the ring $\mathbb{C}[T]$, the parameter $\lambda\in\mathbb{C}[T]$ is some non-constant polynomial and $0\neq\xi\in\mathbb{C}$. It is a function field analogue of the family solved by Mignotte, Peth\H{o} and Roth in the integer case. A feature of our proof is that we avoid the use of height bounds by considering a smaller relevant ring for which we can determine the units more easily. Because of this, the proof is short and the arguments are very elementary (in particular compared to previous results on parametrized Thue equations over function fields).