Integrable Sigma Models and Universal Root $T\bar{T}$ Deformation via Courant-Hilbert Approach

We develop a unified Courant--Hilbert framework for constructing two-dimensional integrable sigma models deformed by two couplings: a marginal one $\gamma$ and an irrelevant one $\lambda$. The integrability condition is encoded in a nonlinear partial differential equation (PDE) for two invariants $(P_1, P_2)$, whose general solution could be expressed through an arbitrary generating function $\ell(\tau)$. This formulation encompasses and extends known models, such as ModMax and Born-Infeld, while introducing new classes of solvable models with closed-form Lagrangians, including those with logarithmic and $q$-deformations. All resulting theories obey a universal root-$T\overline{T}$ flow equation, consistent under dimensional reduction from four-dimensional duality-invariant electrodynamics. Using perturbative expansions, we recover ModMax in the free limit, determine the $\gamma$-dependence of the coupling functions, and show how different flow equations, including a single-trace form, naturally emerge. Our results reveal deep structural connections between self-duality, integrability, and deformation dynamics across different dimensions.