Using an analytic analysis, we derive and solve the constraint equations obtained by folding the 2-spheres which are permuted under the outer automorphism groups of the toric graphs of affine ADE geometries in type II strings. We show that there are classes of solutions depending on the actions of on the elliptic curve E on which flat ADE bundles are fibred. We give two classes of four-dimensional N = 2 superconformal field theory solutions for affine BCFG geometries; one having the right space dimension but exhibiting mirror potentials with branch cuts and a second class sharing the main features with the solutions given by Mayr.