  We present a method for computing all the symmetries of a rational ruled
surface defined by a rational parametrization which works directly in
parametric rational form, i.e. without computing or making use of the implicit
equation of the surface. The method proceeds by translating the problem into
the parameter space, and relies on polynomial system solving. If we want all
the symmetries of the surface, including rotational symmetries, we need to deal
with polynomial systems in four variables; if we are only interested in
involutions (e.g. central symmetries, axial symmetries, reflections in a
plane), we can come down to bivariate polynomial systems. An application to
compute symmetries of an implicit algebraic surface under certain conditions is
also provided.
