Thomas's family of Thue equations over function fields

We consider a function field analogue of Thomas's family of Thue equations X3 − (λ − 1)X2Y − (λ + 2)X Y2 − Y3 = ξ, where the solutions X,Y come from the ring C[T], the parameter λ ∈ C[T] is some non-constant polynomial and 0 ≠ ξ ∈ C. In this paper we solve this family completely.