  We study Yang-Baxter deformations of the $AdS_5 \times S^5$ 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 $AdS_5 \times S^5$ 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 $AdS_5 \times S^5$
after an appropriate coordinate transformation, but this seems not to be
generally the case.
