  Yang-Baxter sigma models, proposed by Klimcik and Delduc-Magro-Vicedo, have
been recognized as a powerful framework for studying integrable deformations of
two-dimensional non-linear sigma models. In this short article, as an important
generalization, we review a non-integrable sigma model in the Yang-Baxter sigma
model approach based on [arXiv:1406.2249]. In particular, we discuss a family
of deformations of the 5D Sasaki-Einstein manifold $T^{1,1}$, instead of the
standard deformations of the $5$-sphere S$^5$. For this purpose, we first
describe a novel construction of $T^{1,1}$ as a supercoset, and provide a
physical interpretation of this construction from viewpoint of the dual
Klebanov-Witten field theory. Secondly, we consider a $3$-parameter deformation
of $T^{1,1}$ by using classical $r$-matrices satisfying the classical
Yang--Baxter equation (CYBE). The resulting metric and NS-NS two-form
completely agree with the ones previously obtained via TsT (T-dual -- shift --
T-dual) transformations, and contain the Lunin-Maldacena background as a
special case. Our result indicates that what we refer to as the
gravity/CYBE(Classical Yang-Baxter Equation) correspondence can be extended
beyond integrable cosets.
