  The work is devoted to gradient Gibbs measures (GGMs) of a SOS model with
countable set $\mathbb Z$ of spin values and having alternating magnetism on
Cayley trees. This model is defined by a nearest-neighbor gradient interaction
potential. Using K\"ulske-Schriever argument, based on boundary law equations,
we give several $q$-height-periodic translations invariant GGMs for $q=2,3,4$.
