We consider the following system of recurrence relations for $n \in \mathbb Z$ and $\vert \lambda \vert=1$ with $\lambda \in \mathbb{C}$

$$a_{n+1} = \lambda a_{n+1}+ \lambda^{-1} a_n + \lambda^{-1} n b_n $$ $$b_{n+1} = \lambda^* b_{n+1}+ (\lambda^*)^{-1} a_n - (\lambda^*)^{-1} n a_n. $$

Here, $\lambda^*$ is the complex conjugate of $\lambda.$

I am interested in initial conditions under which $a_n,b_n$ tend to zero for $n \rightarrow \pm \infty.$