We introduce two classes of homogeneous polynomials and show their role in constructing of integrable hierarchies for some integrable lattices.

We show a broad class of constraints compatible with Itoh-Narita-Bogoyavlenskii lattice hierarchy. All these constraints can be written in the form of discrete conservation law I i+1 = I i with appropriate homogeneous polynomial discrete function I = I[a].

We introduce two classes of discrete polynomials and use them for constructing ordinary difference equations admitting a Lax representation in terms of these polynomials. We also construct lattice integrable hierarchies in their explicit form and show some examples.

We show that by Miura-type transformation the Itoh-Narita-Bogoyavlenskii lattice, for any n ≥ 1, is related to some differential-difference (modified) equation. We present corresponding integrable hierarchies in its explicit form. We study the elementary Darboux transformation for modified equations.

Invariant submanifolds of the so-called Darboux-KP chain [6] are investigated. It is shown that restriction of dynamics on some class of invariant submanifolds yields the extension of the discrete KP hierarchy while the intersections of the latter lead to Lax pairs for a broad class of differential-difference systems with finite number of fields. Some attention is given to investigation of self-similar reductions. It is shown that self-similar ansatzes lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with respect to these parameters. Some examples are provided. In particular it is shown that well known discrete first Painlevé equation (dPI) corresponds to Volterra lattice hierarchy. It is written down equations which naturally generalize dPI in the sense that they have first Painlevé transcedent in continuous limit.