  Classically integrable $\sigma$-models are known to be solutions of the
1-loop RG equations, or "Ricci flow", with only a few couplings running. In
some of the simplest examples of integrable deformations we find that in order
to preserve this property at 2 (and higher) loops the classical $\sigma$-model
should be corrected by quantum counterterms. The pattern is similar to that of
effective $\sigma$-models associated to gauged WZW theories. We consider in
detail 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 required
in order for non-abelian duality to commute with RG flow beyond the 1-loop
order.
