Generalized $T\bar{T}$-like flows for scalar theories in two dimensions

We demonstrate that the necessary condition for $SO(N) \times SO(N)$ duality invariance manifests as a partial differential equation in two-dimensional scalar theories. This condition, expressed as a partial differential equation, corresponds precisely to the integrability condition. We derive a general perturbation solution to this partial differential equation, which includes both a root $T\bar{T}$ flow equation and an irrelevant $T\bar{T}$-like flow equation. Additionally, we identify a general form for these flow equations that commute with each other. Our results establish a general integrable theory characterized by theory-dependent coefficients at each order in the $\lambda$-expansion. This unified framework systematically classifies all integrable theories possessing two Lorentz-invariant variables ($P_1$, $P_2$) while accommodating arbitrary orders of the coupling constants ($\lambda$, $\gamma$). The theory provides a comprehensive classification scheme that encompasses both known and novel integrable systems within this class.