  We present a simple, new method for the 1-loop renormalization of integrable
$\sigma$-models. By treating equations of motion and Bianchi identities on an
equal footing, we derive 'universal' formulae for the 1-loop on-shell
divergences, generalizing case-by-case computations in the literature. Given a
choice of poles for the classical Lax connection, the divergences take a
theory-independent form in terms of the Lax currents (the residues of the
poles), assuming a 'completeness' condition on the zero-curvature equations. We
compute these divergences for a large class of theories with simple poles in
the Lax connection. We also show that $Z_T$ coset models of 'pure-spinor' type
and their recently constructed $\eta$- and $\lambda$-deformations are 1-loop
renormalizable, and 1-loop scale-invariant when the Killing form vanishes.
