  Let $F_n$ denote the $n$-th Fibonacci number and $L_n$ the $n$-th Lucas
number. We completely solve the family of cubic Thue equations
$${(X-F_nY)(X-L_nY)X-Y^3=\pm1}$$ and show that there are no non-trivial
solutions for $n\neq 1,3$.
