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Classically integrable $\sigma$-models are known to be solutions of the1-loop RG equations, or "Ricci flow", with only a few couplings running. Insome of the simplest examples of integrable deformations we find that in orderto preserve this property at 2 (and higher) loops the classical $\sigma$-modelshould be corrected by quantum counterterms. The pattern is similar to that ofeffective $\sigma$-models associated to gauged WZW theories. We consider indetail the examples of the $\eta$-deformation of $S^2$ ("sausage model") and$H^2$, as well as the closely related $\lambda$-deformation of the$SO(1,2)/SO(2)$ coset. We also point out that similar counterterms are requiredin order for non-abelian duality to commute with RG flow beyond the 1-looporder.

Integrable 2d sigma models: quantum corrections to geometry from RG flow