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Yang-Baxter string sigma-models provide a systematic way to deform coset geometries, such as $AdS_{p}\times\ S^{P}$, while retaining the $\sigma$-model integrability. It has been shown that the Yang-Baxter deformation in target space is simply an open-closed string map that can be defined for any geometry, not just coset spaces. Given a geometry with an isometry group and a bivector that is assumed to be a linear combination of an antisymmetric product of Killing vectors, we show the equations of motion of (generalized) supergravity reduce to the Classical Yang-Baxter Equation associated with the isometry group, proving the statement made in [1]. These results bring us closer to the proof of "YB solution generating technique" for (generalized) supergravity advertised in [1] and in particular, provides a more economic way to perform TsT transformations.
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