$(k_0^{(m)})$-periodic Gibbs measures for the fertile three-state Hard-Core model in the case Wand

We consider fertile three-state Hard-Core (HC) models with the activity parameter $\lambda>0$ on a Cayley tree. It is known that there exist four types of such models: wrench, wand, hinge, and pipe. We study $(k_0^{(m)})$-periodic Gibbs measures for the considered model in the case wand on the Cayley tree. Such measures are constructed by means of periodic Gibbs measures. In the case wand on the Cayley tree of order $k=3,4$ for $k_0=2$ and $m=1$ the existence of one ($k_0^{(m)}$)-periodic Gibbs measure proved. Moreover, conditions for the existence and non-uniqueness of ($k_0^{(m)}$)-periodic Gibbs measures on the Cayley tree of order $k=5$ were found.