5. Let $k$ be an integer greater than 1, and the sequence $\left\{a_{n}\right\}$ is defined as follows:
$$
\begin{array}{l}
a_{0}=0, a_{1}=1, \\
a_{n+1}=k a_{n}+a_{n-1}(n=1,2, \cdots) .
\end{array}
$$

Find all $k$ that satisfy the following condition: there exist non-negative integers $l, m (l \neq m)$, and positive integers $p, q$, such that
$a_{l}+k a_{0}=a_{m}+k a_{q^{*}} \quad$ (Xiong Bin)