  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.
