Help with finding a subsequence of a recursive sequence

Question

Given a sequence of non negative integers $(a_n)_{n\geq 1}$, I have a recursive sequence defined by the equation \begin{equation} x_n=a_nx_{n-1}+\lambda x_{n-2}\tag{1}\label{a} \end{equation}

for $n\geq 3$, with initial conditions $x_0,x_1\in\mathbb{R}$ and $\lambda>0$. Because I'm null with the topic "difference equations" I put my first question:

1. If we don't have a closed form for $a_n$: Is there some method to solve the equation \eqref{a}?

On the other hand, suppose that we know that there exists a subsequence $(a_{n_k})_{k\geq 1}$ of $a_n$, which is constant and equal to $l$. So, in this case:

2. Is it possible to solve $x_{n_k}$ from \eqref{a}?

Could anyone help me with those questions or give some references?