  For a rotating dust with a 3-dimensional symmetry group all possible metric
forms can be classified and, within each class, explicitly written out. This is
made possible by the formalism of Pleba\'nski based on the Darboux theorem. In
the resulting coordinates, the Killing vector fields (if any exist) assume a
special form. Each Killing vector field may be either spanned on the fields of
velocity and rotation or linearly independent of them. By considering all such
cases one arrives at the classification. With respect to the structures of the
groups, this is just the Bianchi classification, but with all possible
orientations of the orbits taken into account. In this paper, which is part 1
of a 3-part series, all solutions are considered for which two Killing fields
are spanned on velocity and rotation. The solutions of Lanczos and G\"{o}del
are identified as special cases, and their new invariant definitions are
provided. In addition, a new invariant definition is given of the Ozsvath class
III solution.

General Relativity and Quantum Cosmology