Integrable sigma models and universal root $$T\overline{T }$$ deformation via Courant-Hilbert approach

A bstract We develop a unified Courant-Hilbert framework for constructing two-dimensional integrable sigma models deformed by two couplings: a marginal one γ and an irrelevant one λ . 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 ℓ ( τ ). 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 γ -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.