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- TT 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.
Babaei-Aghbolagh et al. (Fri,) studied this question.