SUSY S-matrix Bootstrap and Friends

We consider the 2D S-matrix bootstrap in the presence of supersymmetry, ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$ symmetry. At the boundary of the allowed S-matrix space we encounter well known integrable models such as the supersymmetric sine-Gordon and restricted sine-Gordon models, novel elliptic deformations thereof, as well as a two parameter family of ${\mathbb{Z}}_4$ elliptic S-matrices previously proposed by Zamolodchikov. We highlight an intricate web of relations between these various S-matrices.