Certain mathematical aspects of the static pion-nucleon theory are investigated. We start from the fact that the theory in its uniformized form (cuts transformed away) leads to a system of functional equations for the S-matrix. The nonlinear mapping involved in the functional equations is a second-order Cremona transformation. After a summary of the general properties of Cremona transformations, the special transformations are studied which arise in the symmetric scalar and the Chew-Low theory respectively. The emphasis is on the possibility to separate, by means of a finite-order Cremona transformation, the functional equations into a set of uncoupled ones. For the symmetric scalar theory, the separation is trivial. For the Chew-Low theory, a proof of nonseparability is given.