Gradient Gibbs measures of a SOS model on Cayley trees: 4-periodic boundary laws

For SOS (solid-on-solid) model with external field and with spin values from the set of all integers, on a Cayley tree we give gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (a function defined on vertices of Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs which correspond to periodic boundary laws.