The analytic properties of the amplitudes satisfying Nagaoka's and Suhl's integral equations, respectively, are examined without solving the equations. The technique used is to analytically continue the equations themselves by deforming the contours of integration appearing in them. This turns out to be a very simple method of examining the singularity structure of the amplitudes on the unphysical sheet. The two sets of equations have extremely similar analytic properties. In fact, they are so similar that we are led to conjecture a possible exact solution of Nagaoka's equation analogous to Suhl's result. The solution is similar to Hamann's approximate solution, but has no cuts. We prove that the exact solution cannot have cuts. It seems possible that the techniques developed in this paper will prove useful in examining more exact theories incorporating higher-order effects.