  We study invariant surfaces generated by one-parameter subgroups of simply
and pseudo isotropic rigid motions. Basically, the simply and pseudo isotropic
geometries are the study of a three-dimensional space equipped with a rank 2
metric of index zero and one, respectively. We show that the one-parameter
subgroups of isotropic rigid motions lead to seven types of invariant surfaces,
which then generalizes the study of revolution and helicoidal surfaces in
Euclidean and Lorentzian spaces to the context of singular metrics. After
computing the two fundamental forms of these surfaces and their Gaussian and
mean curvatures, we solve the corresponding problem of prescribed curvature for
invariant surfaces whose generating curves lie on a plane containing the
degenerated direction.
