There should be a standard argument for the following claim: If the system of equations of the form $$a_{n-1}u_{n-1}+(b_{n}-z)u_{n}+a_{n}u_{n+1}=0, \quad n\in\mathbb{Z},$$ where $a_{n}\in\mathbb{C}\setminus\{0\}$ and $b_{n}\in\mathbb{C}$, has two linearly independent square summable solutions (i.e., in $\ell^{2}(\mathbb{Z})$) for one $z\in\mathbb{C}$, then the same holds true for all $z\in\mathbb{C}$.
Is there a reasonably simple (or more or less elementary) proof?