Description of fixed points of an infinite dimensional operator

We consider an infinite-dimensional non-linear operator related to a hard core (HC) model with a countable set $\mathbb{N}$ of spin values. It is known that finding the fixed points of an infinite-dimensional operator is generally impossible. But we have fully analyzed the fixed points of an infinite-dimensional operator by applying a technique of reducing an infinite-dimensional operator to a two-dimensional operator. The set of parameters is divided into subsets $A_{i,j},$ where the index $i$ means the number of fixed points on the line $y=x$, $j$ means the number of fixed points outside of $y=x\mathrm{.}$ The number of fixed points can be up to seven, and the explicit form of each fixed point is found.