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