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We study Yang-Baxter deformations of the AdS_{5}×S^{5} superstring with non-Abelian classical r-matrices which satisfy the homogeneous classical Yang-Baxter equation (CYBE). By performing a supercoset construction, we can get deformed AdS_{5}×S^{5} backgrounds. While this is a new area of research, the current understanding is that Abelian classical r-matrices give rise to solutions of type IIB supergravity, while non-Abelian classical r-matrices lead to solutions of the generalized supergravity equations. We examine here some examples of non-Abelian classical r-matrices and derive the associated backgrounds explicitly. All of the resulting backgrounds satisfy the generalized equations. For some of them, we derive "T-dualized" backgrounds by adding a linear coordinate dependence to the dilaton and show that these satisfy the usual type IIB supergravity equations. Remarkably, some of the "T-dualized" backgrounds are locally identical to undeformed AdS_{5}×S^{5} after an appropriate coordinate transformation, but this seems not to be generally the case.