Integrable deformed H$_{_{4}}$ WZW models and their non-Abelian duals as solutions of generalized supergravity equations

We show that the Yang-Baxter (YB) deformed backgrounds of the Wess-Zumino-Witten (WZW) model based on the $H_{_{4}}$ Lie group can be considered as solutions of the generalized supergravity equations (GSEs). Then, by applying the Poisson-Lie T-duality in the presence of spectator fields, we obtain the non-Abelian target space duals of those models. It is shown that all dual models (except for one model) are integrable and most interestingly, they satisfy the GSEs. As a final remark, we show that all solutions of the GSEs for the original models are trivial.