  We further study integrable deformations of the AdS$_5\times$S$^5$
superstring by following the Yang-Baxter sigma model approach with classical
$r$-matrices satisfying the classical Yang-Baxter equation (CYBE). Deformed
string backgrounds specified by $r$-matrices are considered as solutions of
type IIB supergravity, and therefore the relation between gravitational
solutions and $r$-matrices may be called the gravity/CYBE correspondence. In
this paper, we present a family of string backgrounds associated with a
classical $r$-matrices carrying two parameters and its three-parameter
generalization. The two-parameter case leads to the metric and NS-NS two-form
of a solution found by Hubeny-Rangamani-Ross [hep-th/0504034] and another
solution in [arXiv:1402.6147]. For all of the backgrounds associated with the
three-parameter case, the metric and NS-NS two-form are reproduced by
performing TsT transformations and S-dualities for the undeformed
AdS$_5\times$S$^5$ background. As a result, one can anticipate the R-R sector
that should be reproduced via a supercoset construction.
