Ground states of Ising-Potts model on Cayley tree

It is known that for low temperatures, a ground state is associated with a limiting Gibbs measure. This is why, the studying of the sets of ground states for a given physical system is a topical issue.
We consider a model of mixed type on the Cayley tree, which is referred to as Ising-Potts model, that is, the Ising and Potts models are related with the parameter $\alpha$, where $\alpha \in [0,1]$. In the paper we study the ground state for the Ising-Potts model with three states on the Cayley tree. It is known that there exists a one-to-one correspondence between the set of the vertices $V$ of the Cayley tree of order $k$ and a group $G_k$ being a free product of $k+1$ cyclic groups of second order. We define periodic and weakly periodic ground states corresponding to normal divisors of the group $G_k$. For the Ising-Potts model we describe the set of periodic and weakly periodic ground states corresponding to normal divisors of index $2$ of the group $G_k$. We prove that for some values of the parameters there exist no such periodic (non translation-invariant) ground states. We also prove that for a normal subgroup consisting of even layers there exist periodic (non translation-invariant) ground states and we also prove the existence of weakly-periodic (non periodic) ground states.