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In the first half of the paper, we study in details NS-branes, including theNS5-brane, the Kaluza-Klein monopole and the exotic $5_2^2$- or Q-brane,together with Bianchi identities for NSNS (non)-geometric fluxes.Four-dimensional Bianchi identities are generalized to ten dimensions withnon-constant fluxes, and get corrected by a source term in presence of anNS-brane. The latter allows them to reduce to the expected Poisson equation.Without sources, our Bianchi identities are also recovered by squaring anilpotent $Spin(D,D) \times \mathbb{R}^+$ Dirac operator. Generalized Geometryallows us in addition to express the equations of motion explicitly in terms offluxes. In the second half, we perform a general analysis of ten-dimensionalgeometric backgrounds with non-geometric fluxes, in the context of$\beta$-supergravity. We determine a well-defined class of such vacua, that arenon-geometric in standard supergravity: they involve $\beta$-transforms, amanifest symmetry of $\beta$-supergravity with isometries. We show as well thatthese vacua belong to a geometric T-duality orbit.

NS-branes, source corrected Bianchi identities, and more on backgrounds with non-geometric fluxes