  We present a general construction of integrable degenerate $\mathcal
E$-models on a 2d manifold $\Sigma$ using the formalism of Costello and
Yamazaki based on 4d Chern-Simons theory on $\Sigma \times \mathbb{C}P^1$. We
begin with a physically motivated review of the mathematical results of
[arXiv:2008.01829] where a unifying 2d action was obtained from 4d Chern-Simons
theory which depends on a pair of 2d fields $h$ and $\mathcal L$ on $\Sigma$
subject to a constraint and with $\mathcal L$ depending rationally on the
complex coordinate on $\mathbb{C}P^1$. When the meromorphic 1-form $\omega$
entering the action of 4d Chern-Simons theory is required to have a double pole
at infinity, the constraint between $h$ and $\mathcal L$ was solved in
[arXiv:2011.13809] to obtain integrable non-degenerate $\mathcal E$-models. We
extend the latter approach to the most general setting of an arbitrary 1-form
$\omega$ and obtain integrable degenerate $\mathcal E$-models. To illustrate
the procedure we reproduce two well known examples of integrable degenerate
$\mathcal E$-models: the pseudo dual of the principal chiral model and the
bi-Yang-Baxter $\sigma$-model.
