Translation-invariant Gibbs measures for fertile three-state “hard core” models on a Cayley tree

We consider fertile three-state "hard core" models with the activity parameter $\lambda>0$ on an order-three Cayley tree. It is known that there exist four types of such models: in two of them, the translation-invariant Gibbs measure is unique for $\lambda>0$, and in the other two, a value $\lambda_\mathrm{cr}$ is found such that there exist only three translation-invariant Gibbs measures for $\lambda>\lambda_\mathrm{cr}$ and a single translation-invariant Gibbs measure for $\lambda\le\lambda_\mathrm{cr}$.