  We consider $\sigma$-models on para-complex $\mathbb{Z}_T$-cosets, which are
analogues of those on complex homogeneous target spaces considered recently by
D. Bykov. For these models, we show the existence of a gauge-invariant Lax
connection whose Poisson brackets are ultralocal. Furthermore, its light-cone
components commute with one another in the sense of Poisson brackets. This
extends a result of O. Brodbeck and M. Zagermann obtained twenty years ago for
hermitian symmetric spaces.
