The P surface of Hermann Amandus Schwarz has a cubic period lattice. Let’s rotate the surface by 45 degrees so that the fundamental cube balances on an edge. The resulting surface can still being cut by vertical symmetry planes and horizontal straight lines into a simply connected minimal polygon, and in this description it possesses an explicit 2-parameter deformation. The period lattice is then a base-centered orthorhombic Bravais lattice:
Rotating the surface about the (now) vertical axis produces a rectangular quotient torus on which the divisor of the square of the Gauß map is given by the figure below, with a+b=1/2, Re a = Re c and Re b = Re d.
This deformation was already known to Schwarz as follows: The eight branched values of the Gauß map lie on the vertices of two axes-symmetrical, spherical rectangles in the xz- and yz-plane, respectively. You can locate the branched points on the surface by looking for the intersections between straight lines and planar symmetry curves.
Possible limits include noded planes, doubly periodic Scherk surfaces, and KMR surfaces.