Anisotropic Ising model with countable set of spin values on Cayley tree

In this paper we investigate of an infinite system of functional equations for the Ising model with competing interactions and countable spin values $0,1,\ldots$ and non zero filed on a Cayley tree of order two. We derived an infinite system of functional equations for the Ising model that is we describe conditions on $h_x$ guaranteeing compatibility of distributions $\mu^{(n)}(\sigma_n)$.