  Two-dimensional $\sigma$-models with $\mathbb{Z}_N$-symmetric homogeneous
target spaces have been shown to be classically integrable when introducing
WZ-terms in a particular way. This article continues the search for new models
of this type now allowing some kinetic terms to be absent, analogously to the
Green-Schwarz superstring $\sigma$-model on $\mathbb{Z}_4$-symmetric
homogeneous spaces. A list of such integrable $\mathbb{Z}_N$-symmetric
(super)coset $\sigma$-models for $N \leq 6$ and their Lax pairs is presented.
For arbitrary $N$, a big class of integrable models is constructed that
includes both the known pure spinor and Green-Schwarz superstring on
$\mathbb{Z}_4$-symmetric cosets.
  Integrable Yang-Baxter deformations of this class of $\mathbb{Z}_N$-symmetric
(super)coset $\sigma$-models can be constructed in same way as in the known
$\mathbb{Z}_2$- or $\mathbb{Z}_4$-cases. Deformations based on solutions of the
modified classical Yang-Baxter equation, the so-called $\eta$-deformation,
require deformation of the constants defining the Lagrangian and the
corresponding Lax pair. Homogeneous Yang-Baxter deformations (i.e. those based
on solutions to the classical Yang-Baxter equation) leave the equations of
motion and consequently the Lax pair invariant and are expected to be
classically equivalent to the undeformed model.
  As an example, the relationship between $\mathbb{Z}_3$-symmetric homogeneous
spaces and nearly (para-)K\"ahler geometries is revisited. Confirming existing
literature it is shown that the integrable choice of WZ-term in the
$\mathbb{Z}_3$-symmetric coset $\sigma$-model associated to a nearly K\"ahler
background gives an imaginary contribution to the action.
