  We explicitly derive Lax pairs for string theories on Yang-Baxter deformed
backgrounds, 1) gravity duals for noncommutative gauge theories, 2)
$\gamma$-deformations of S$^5$, 3) Schr\"odinger spacetimes and 4) abelian
twists of the global AdS$_5$\,. Then we can find out a concise derivation of
Lax pairs based on simple replacement rules. Furthermore, each of the above
deformations can be reinterpreted as a twisted periodic boundary conditions
with the undeformed background by using the rules. As another derivation, the
Lax pair for gravity duals for noncommutative gauge theories is reproduced from
the one for a $q$-deformed AdS$_5\times$S$^5$ by taking a scaling limit.
