  We consider an SOS (solid-on-solid) model, with spin values from the set of
all integers, on a Cayley tree of order k and are interested in
translation-invariant gradient Gibbs measures (GGMs) of the model. Such a
measure corresponds to a boundary law (a function defined on vertices of the
Cayley tree) satisfying a functional equation. In the ferromagnetic SOS case on
the binary tree we find up to five solutions to a class of 4-periodic boundary
law equations (in particular, some 2-periodic ones). We show that these
boundary laws define up to four distinct GGMs. Moreover, we construct some
3-periodic boundary laws on the Cayley tree of arbitrary order k, which define
GGMs different from the 4-periodic ones.
