  By addition of non-zero, but torsionless $B$-field, we expand the
classification of (non-)Abelian T-duals of the flat background in four
dimensions with respect to one-, two-, three-, and four-dimensional subgroups
of Poincar\'e group. We discuss the influence of the additional $B$-field on
the process of dualization and identify essential parts of the torsionless
$B$-field that cannot be eliminated in general by coordinate or gauge
transformation of the dual background. These effects are demonstrated using
particular examples. Due to their physical importance, we focus on duals whose
metrics represent plane-parallel waves. Besides the previously found metrics,
we find new pp-waves depending on parameters originating from the torsionless
$B$-field. These pp-waves are brought into their standard forms in Brinkmann
and Rosen coordinates.
