  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.
