Surfaces in E3 invariant under a one parameter group of isometries of E3

We develop a convenient surface theory in E3 in order to apply it to the class of the surfaces invariant under a one-parameter group of isometries of E3. In this way we derive intrinsic characterizations along with several results of subclasses of this class of surfaces that satisfy certain preassigned properties. In the process all results are also effortlessly derived. Among these subclasses are those with surfaces; of constant mean curvature, of constant Gaussian curvature, isothermic, with constant difference or ratio of the principal curvatures.