  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
