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For a rotating dust with a 3-dimensional symmetry group all possible metricforms can be classified and, within each class, explicitly written out. This ismade possible by the formalism of Pleba\'nski based on the Darboux theorem. Inthe resulting coordinates, the Killing vector fields (if any exist) assume aspecial form. Each Killing vector field may be either spanned on the fields ofvelocity and rotation or linearly independent of them. By considering all suchcases one arrives at the classification. With respect to the structures of thegroups, this is just the Bianchi classification, but with all possibleorientations of the orbits taken into account. In this paper, which is part 1of a 3-part series, all solutions are considered for which two Killing fieldsare spanned on velocity and rotation. The solutions of Lanczos and G\"{o}delare identified as special cases, and their new invariant definitions areprovided. In addition, a new invariant definition is given of the Ozsvath classIII solution.Comment: 23 pages, LaTe

Rotating dust solutions of Einstein’s equations with 3-dimensional symmetry groups. I. Two Killing fields spanned on uα and wα