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Abstract
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Generalized supergravity equations (GSEs) were originally proposed as a modification of the standard IIB supergravity equations to
satisfy the background of $\eta$-deformed $AdS_5 \times S^5$ introduced by Arutyunov {\it et al}. In this study, we proceed to write down the
GSEs for the Wess-Zumino-Witten (WZW) models based on Lie groups.
First, we simplify the GSEs for the WZW model by imposing the conditions for vanishing of the one-loop beta function equations.
Then, by introducing an {\it Ansatz} for the Killing vector field $I$, it is shown that the Killing equation ${\cal L}_{_{I}} G_{_{\mu \nu}}=0$ is held.
In addition, we introduce a generalized Killing vector such that the existence of a solution to the GSEs requires that this vector be light-like.
In this way, we solve the simplified GSEs for the WZW models constructed on Lie groups up to dimension five.
Unfortunately, two of the groups do not have light-like vectors and therefore do not admit the GSEs.
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