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

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 $\widehat G$ of the group realizing the Cayley tree, we study $\widehat hG$-periodic Gibbs measures. The existence of an uncountable set of $\widehat hG$-periodic Gibbs measures (which are not translation invariant and not “checkerboard” periodic) is proved.