Gradient Gibbs measures of an SOS model with alternating magnetism on Cayley trees

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 $q=2,3,4$.