Dynamically restoring conformal invariance in (integrable) $\sigma$-models

Integrable $\lambda$-deformed $\sigma$-models are characterized by an underlying current algebra/coset model CFT deformed, at the infinitesimal level, current/parafermion bilinears. We promote deformation parameters to dynamical functions of time introduced as extra coordinate. It is conceivable that appropriately constraining them, beta-functions vanish and consequently $\sigma$-model stays conformal. Remarkably, we explicitly materialize this scenario in several cases having a single even multiple parameters. These generically obey system non-linear second-order ordinary differential equations. They solved fixed points RG flow original $\sigma$-model. Moreover, choosing initial conditions may interpolate between varies from far past future.Finally, present extension our analysis Yang--Baxter deformed PCMs.