  We study the renormalisation of a large class of integrable $\sigma$-models
obtained in the framework of affine Gaudin models. They are characterised by a
simple Lie algebra $\mathfrak{g}$ and a rational twist function $\varphi(z)$
with simple zeros, a double pole at infinity but otherwise no further
restrictions on the pole structure. The crucial tool used in our analysis is
the interpretation of these integrable theories as $\mathcal{E}$-models, which
are $\sigma$-models studied in the context of Poisson-Lie T-duality and which
are known to be at least one- and two-loop renormalisable. The moduli space of
$\mathcal{E}$-models still contains many non-integrable theories. We identify
the submanifold formed by affine Gaudin models and relate its tangent space to
curious matrices and semi-magic squares. In particular, these results provide a
criteria for the stability of these integrable models under the RG-flow. At one
loop, we show that this criteria is satisfied and derive a very simple
expression for the RG-flow of the twist function, proving a conjecture made
earlier in the literature.
