Affine equivalences of rational surfaces of translation, and applications to rational minimal surfaces

We provide an algorithm for determining whether two rational surfaces of translation are affinely equivalent. In turn, this also provides an algorithm for determining whether two rational minimal surfaces are affinely equivalent. This algorithm is applied to determine the symmetries of rational minimal surfaces, in particular the higher-order Enneper surfaces. Finally certain parity-like conditions in the Weierstrass form of minimal surfaces are used to construct minimal surfaces with prescribed symmetries.