Continuant Diophantine equations

We investigate a family of Diophantine polynomial equations which involve continuant functions. In particular, given a polynomial $P(x)\in \mathbb{Z}[x]$ and $n\in \mathbb{N}$, we consider the equation $P(K_n(x_1,\ldots, x_n)) = K_{n+1}(x_0,\ldots,x_n)K_{n+1}(x_1,\ldots, x_{n+1})$. We show that with certain restrictions on $P(x)$ the set of its solutions has a rich structure. In particular, we provide several ways of generating new solutions from the existing ones. In the last section we discuss the relation between the solutions of the above Diophantine equation for arbitrary values of $n$ and factorisations $P(m) = d_1d_2$ for integers $m,d_1$ and $d_2$.