Gibbs measures for a generalized Potts model with the interaction radius two on a Cayley tree

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We study a generalized Potts model on a Cayley tree of order k = 3. Under some conditions on the parameters, we show that there exist at most two translation-invariant Gibbs measures and a continuum of Gibbs measures that are not translation invariant. For any index-two normal divisor Ĝ of the group realizing the Cayley tree, we study Ĝ-periodic Gibbs measures. The existence of an uncountable set of Ĝ-periodic Gibbs measures (which are not translation invariant and not “checkerboard” periodic) is proved.