  A regular parallelism of real projective 3-space PG(3,R) is an equivalence
relation on the line space such that every class is equivalent to the set of
1-dimensional complex subspaces of a 2-dimensional complex vector space. We
shall assume that the set of classes is compact, and characterize those regular
parallelisms that admit an action of a 2-dimensional torus group. We prove that
there is a one-dimensional subtorus fixing every parallel class. From this
property alone we deduce that the parallelism is a 2- or 3-dimensional regular
parallelism in the sense of Betten and Riesinger. If a 2-torus acts, then the
parallelism can be described using a so-called generalized line star which
admits a 1-torus action. We also study examples of such parallelisms by
constructing generalized line stars. In particular, we prove a claim which was
presented by Betten and Riesinger with an incorrect proof. The present article
continues a series of papers by the first author on parallelisms with large
groups.
