Functional equations for the Potts model with competing interactions on a Cayley tree

In this paper, we consider an infinite system of functional equations for the Potts model with competing interactions of radius $r=2$ and countable spin values $0,1,\dots$, and non-zero-filled, on a Cayley tree of order two. We describe conditions on $h_x$ guaranteeing compatibility of distributions $\mu^{(n)}(\sigma_n)$.