On integrals for some class of ordinary difference equations admitting a Lax pair representation

We consider two infinite classes of ordinary difference equations admitting Lax pair representation. Discrete equations in these classes are parameterized by two integers $k\geq 0$ and $s\geq k+1$. We describe the first integrals for these two classes in terms of special discrete polynomials. We show an equivalence of two difference equations belonged to different classes corresponding to the same pair $(k, s)$. We show that solution spaces $\mathcal{N}^k_s$ of different ordinary difference equations with fixed value of $s+k$ are organized in chain of inclusions.