Height-periodic gradient Gibbs measures for generalised SOS model on Cayley tree

For generalised SOS model with spin values from countable set on a Cayley tree we consider gradient Gibbs measures(GGMs). Such a measure corresponds to a boundary law (i.e. an infinite dimensional vector-valued function defined on vertices of the Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs of boundary laws which are independent from vertices of the Cayley tree and have periodic and mirror-symmetric coordinates. Namely, in this paper the particular classes of height-periodic boundary laws of period q≤ 3 are studied and solutions are classified by their period and mirror-symmetry.