  Exact conformal field theories (CFTs) are obtained by using the approach of
Poisson-Lie (PL) T-duality in the presence of spectators. We explicitly
construct some non-Abelian T-dual $\sigma$-models (here as the PL T-duality on
a semi-Abelian double) on $2+2$-dimensional target manifolds $M \approx O
\times \bf G$ and ${\tilde M} \approx O \times {\bf {\tilde G}}$, where $\bf G$
and ${\bf {\tilde G}}$ as two-dimensional real non-Abelian and Abelian Lie
groups act freely on $M$ and $\tilde M$, respectively, while $O$ is the orbit
of $\bf G$ in $M$. The findings of our study show that the original models are
equivalent to Wess-Zumino-Witten (WZW) models based on the Heisenberg $(H_4)$
and $GL(2,\mathbb{R})$ Lie groups. In this way, some new T-dual backgrounds for
these WZW models are obtained. For one of the duals of the $H_4$ WZW model, we
show that the model is self-dual. In the case of the $GL(2,\mathbb{R})$ WZW
model it is observed that the duality transformation changes the asymptotic
behavior of solutions from $AdS_{3} \times \mathbb{R}$ to flat space. Then, the
structure and asymptotic nature of the dual spacetime of this model including
the horizon and singularity are determined. We furthermore get the non-critical
Bianchi type III string cosmological model with a non-vanishing field strength
from T-dualizable $\sigma$-models and show that this model describes an exact
CFT (equivalent to the $GL(2,\mathbb{R})$ WZW model). After that, the conformal
invariance of T-dual models up to two-loop order (first order in $\alpha'$) is
discussed.
