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By addition of non-zero, but torsionless $B$-field, we expand theclassification of (non-)Abelian T-duals of the flat background in fourdimensions with respect to one-, two-, three-, and four-dimensional subgroupsof Poincar\'e group. We discuss the influence of the additional $B$-field onthe process of dualization and identify essential parts of the torsionless$B$-field that cannot be eliminated in general by coordinate or gaugetransformation of the dual background. These effects are demonstrated usingparticular examples. Due to their physical importance, we focus on duals whosemetrics 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 Brinkmannand Rosen coordinates.

Plane-parallel waves as duals of the flat background III: T-duality with torsionless $B$-field