Translation invariant Gibbs measures for three state Hard-Core models in the case Hinge

In this paper, we study the translation-invariant Gibbs measures for the fertile three-state Hard-Core (HC) models with the activity parameter $\lambda_i>0, i=0,1,2$ on a Cayley tree. It is known that there exist four types of such models: "wrench", "wand", "hinge" and "pipe". In the case "hinge" under some conditions on parameters on a Cayley tree of order two, three the exact number of the translation invariant Gibbs measures found.