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Stationary, axisymmetric and asymptotically flat spacetimes of dust of whichtrajectories are integral curves of the time translation Killing vector areinvestigated. The flow has no Newtonian limit. Asymptotic flatness implies theexistence of singularities of the curvature scalar that are distributions andthat are not isolated from regularity regions of the flow. The singularitiesare closely related to the presence of additional stresses that contributenegative active mass to the total (Komar) mass, which is zero forasymptotically flat spacetimes. Several families of solutions are constructedand a multipole expansion is presented.Comment: v2: typos corrected, v3: removed comments on C&T model (updated and  transfered to a separate paper) v4: new class of solutions and new section  added on curvature singularities in asymptotically flat van Stockum-Bonnor  spacetime

Van Stockum-Bonnor spacetimes of rigidly rotating dust