$O(d,d)$ transformations preserve classical integrability

In this note, we study the action of $O(d,d)$ transformations on the integrable structure of two-dimensional non-linear sigma models via the doubled formalism. We construct the Lax pairs associated with the $O(d,d)$-transformed model and find that they are in general non-local because they depend on the winding modes. We conclude that every $O(d,d;\mathbb{R})$ deformation preserves integrability. As an application we compute the Lax pairs for continuous families of deformations, such as $J\bar{J}$ marginal deformations and TsT transformations of the three-sphere with $H$-flux.