  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 $k\geq 2$ and
$\Lambda=\sum_i\lambda_i$. For $\Lambda=+\infty$ there is no
translation-invariant Gibbs measure (TIGM); Let $k=2$ and $\Lambda<+\infty$.
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 $\Lambda<+\infty$. 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.
