  In this paper, we study conformal points among the class of
$\mathcal{E}$-models. The latter are $\sigma$-models formulated in terms of a
current Poisson algebra, whose Lie-theoretic definition allows for a purely
algebraic description of their dynamics and their 1-loop RG-flow. We use these
results to formulate a simple algebraic condition on the defining data of such
a model which ensures its 1-loop conformal invariance and the decoupling of its
observables into two chiral Poisson algebras, describing the classical left-
and right-moving fields of the theory. In the case of so-called non-degenerate
$\mathcal{E}$-models, these chiral sectors form two current algebras and the
model takes the form of a WZW theory once realised as a $\sigma$-model. The
case of degenerate $\mathcal{E}$-models, in which a subalgebra of the current
algebra is gauged, is more involved: the conformal condition yields a wider
class of theories, which includes gauged WZW models but also other examples,
seemingly different, which however sometimes turn out to be related to gauged
WZW models based on other Lie algebras. For this class, we build non-local
chiral fields of parafermionic-type as well as higher-spin local ones, forming
classical $\mathcal{W}$-algebras. In particular, we find an explicit and
efficient algorithm to build these local chiral fields. These results (and
their potential generalisations discussed at the end of the paper) open the way
for the quantisation of a large class of conformal $\mathcal{E}$-models using
the standard operator formalism of two-dimensional CFT.
