Besides the rather canonical deformation of Alan Schoen’s FR-D surface, there is also an exotic version that solves the same period problem as the canonical deformation, but with fewer symmetries. The statements on this page are at this point only based on numerical experiments.
The divisor of G4(z) on the quotient torus relaxes to
with the Abel condition asking for 1/2 + 2 a + 3 b = 3 c + 2 d. Thus the surfaces lose their normal symmetries about horizontal lines.
The period problem becomes 3-dimensional, which is somewhat unpleasant. Besides the established symmetrical deformation, there is at least one other solution family. Below is a graph of solutions of the parameters as functions of a.
This exotic family appears to intersect the standard F-RD surfaces in one surface (but not the one in the rhombic dodecahedron). This behavior is very much like that of Hao Chen’s exotic D-surface.
If true, this would show that the F-RD surface is a singular bifurcation instance in the sense that there are more than two different deformations with the same period lattice.
Both limits of this deformation are identical albeit shifted: We see horizontal planes joined by Wohlgemuth surface nodes. This type of limit has not been employed yet for (re-) construction purposes.
Mathematica Notebook to come