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

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

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

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

Observation: If there is a limit $a_n \rightarrow a$ and $b_n \rightarrow b$, then indeed by the last term in each row, it has to be zero and $a_n =b_n=o(n)$