  The Root-$T \overline{T}$ flow was recently introduced as a universal and
classically marginal deformation of any two-dimensional translation-invariant
field theory. The flow commutes with the (irrelevant) $T \overline{T}$ flow and
it can be integrated explicitly for a large class of actions, leading to
non-analytic Lagrangians reminiscent of the four-dimensional Modified-Maxwell
theory (ModMax). It is not a priori obvious whether the Root-$T \overline{T}$
flow preserves integrability, like it is the case for the $T \overline{T}$
flow. In this paper we demonstrate that this is the case for a large class of
classical models by explicitly constructing a deformed Lax connection. We
discuss the principal chiral model and the non-linear sigma models on symmetric
and semi-symmetric spaces, without or with Wess-Zumino term. We also construct
Lax connections for the two-parameter families of theories deformed by both
Root-$T \overline{T}$ and $T \overline{T}$ for all of these models.
