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We consider Gradient Gibbs measures corresponding to a periodic boundary lawfor a generalized SOS model with spin values from a countable set, on Cayleytrees. On the Cayley tree, detailed information on Gradient Gibbs measures formodels of SOS type are given in \cite{3, 16,8,11} and we continue the works forthe generalized SOS model. Namely, in this paper, the problem of findingGradient Gibbs measures that correspond to periodic boundary laws is reduced toa functional equation and by solving the equation all Gradient Gibbs measureswith 4 periodic boundary laws are found.

Gradient Gibbs measures with periodic boundary laws of a generalized SOS model on a Cayley tree