In general, Faddeev equations need as input a potential that describes the interaction between two individual particles.
It is also possible to introduce a term in the equation in order to take also three-body forces into account.
In nuclear physics, the off the energy shell nucleon-nucleon interaction has been studied by analyzing (n,2n) and (p,2p) reactions on deuterium targets, using the Faddeev Equations.
The Coulomb interaction between two protons is a special problem, in that its expansion in separable potentials does not converge, but this is handled by matching the Faddeev solutions to long range Coulomb solutions, instead of to plane waves.
The physical nucleon-nucleon interaction, which involves exchange of mesons, is not expected to be either local or separable.