Tranlation-invariant Gibbs measures for the Hard-Core model with a countable set of spin values

In this paper, we study the Hard Core (HC) model with a countable set $\mathbb{Z}$ of spin values on a Cayley tree of order $k=2$. This model is defined by a countable set of parameters (that is, the activity function ${\lambda }_i>0$, $i\in \mathbb{Z}$). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained: Let $k\ge 2$ and $\mathrm{\Lambda }={\sum }_i{\lambda }_i$. For $\mathrm{\Lambda }=+\mathrm{\infty }$ there is no translation-invariant Gibbs measure (TIGM); Let $k=2$ and $\mathrm{\Lambda }<+\mathrm{\infty }$. For the model under constraint such that at $G$-admissible graph the loops are imposed at two vertices of the graph, the uniqueness of TIGM is proved; Let $k=2$ and $\mathrm{\Lambda }<+\mathrm{\infty }$. For the model under constraint such that at $G$-admissible graph the loops are imposed at three vertices of the graph, the uniqueness and non-uniqueness conditions of TIGMs are found.