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We give the classification of T-duals of the flat background in fourdimensions with respect to one-, two-, and three-dimensional subgroups of thePoincar\'e group using non-Abelian T-duality with spectators. As duals we findbackgrounds for sigma models in the form of plane-parallel waves ordiagonalizable curved metrics often with torsion. Among others, we find exactlysolvable time-dependent isotropic pp-wave, singular pp-waves, or generalizedplane wave (K-model).

Plane-parallel waves as duals of the flat background II: T-duality with spectators