Alternative Gibbs measure for fertile three-state Hard-Core models on a Cayley tree

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”,. In cases “wand”, and “hinge”\ on a Cayley tree of arbitrary order a complete description of translation-invariant Gibbs measures is obtained. The conception of alternative Gibbs measure is introduced and in the case “wand”, translational invariance conditions for alternative Gibbs measures are found. Also, we show that the existence of alternative Gibbs measures which are not translation-invariant