Supergravity solution-generating techniques and canonical transformations of ?-models from O(D, D)

Within the framework of flux formulation Double Field Theory (DFT) we employ a generalised Scherk-Schwarz ansatz and discuss classification twists that in presence strong constraint give rise to constant fluxes interpreted as gaugings. We analyse various possibilities turning on $H_{ijk}, F_{ij}{}^k, Q_i{}^{jk}$ $R^{ijk}$, solutions for allowed each case. While do not impose DFT (or equivalently supergravity) equations motion, our results provide solution-generating techniques supergravity when applied background does solve equations. At same time, also canonical transformations 2-dimensional $\sigma$-models, fact which is interesting especially because these are integrability-preserving worldsheet. Both $\sigma$-models arise maps leave their flat derivatives invariant. These include known abelian/non-abelian/Poisson-Lie T-duality transformations, Yang-Baxter deformations, well novel generalisations them.