  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 $d_2$.
