  We present further examples of the correspondence between solutions of type
IIB supergravity and classical $r$-matrices satisfying the classical
Yang-Baxter equation (CYBE). In the previous works, classical $r$-matrices have
been composed of generators of only one of either $\mathfrak{so}(2,4)$ or
$\mathfrak{so}(6)$. In this paper, we consider some examples of $r$-matrices
with both of them. The $r$-matrices of this kind contain (generalized)
Schr\"odinger spacetimes and gravity duals of dipole theories. It is known that
the generalized Schr\"odinger spacetimes can also be obtained via a certain
class of TsT transformations called null Melvin twists. The metric and NS-NS
two-form are reproduced by following the Yang-Baxter sigma-model description.
