  The non-Abelian T-dualization of the BTZ black hole is discussed in detail by
using the Poisson-Lie T-duality in the presence of spectators. We explicitly
construct a dual pair of sigma models related by Poisson-Lie symmetry. The
original model is built on a $2+1$-dimensional manifold ${\cal M} \approx O
\times \bf G$, where $\bf G$ as a two-dimensional real non-Abelian Lie group
acts freely on ${\cal M}$, while $O$ is the orbit of $\bf G$ in ${\cal M}$. The
findings of our study show that the original model indeed is canonically
equivalent to the $SL(2,\mathbb{R})$ Wess-Zumino-Witten (WZW) model for a given
value of the background parameters. Moreover, by a convenient coordinate
transformation we show that this model describes a string propagating in a
spacetime with the BTZ black hole metric in such a way that a new family of the
solutions to low energy string theory with the BTZ black hole vacuum metric,
constant dilaton field and a new torsion potential is found. The dual model is
built on a $2+1$-dimensional target manifold $\tilde {\cal M}$ with
two-dimensional real Abelian Lie group ${\tilde {\bf G}}$ acting freely on it.
We further show that the dual model yields a three-dimensional charged black
string for which the mass $M$ and axion charge $Q$ per unit length are
calculated. After that, the structure and asymptotic nature of the dual
space-time including the horizon and singularity are determined.
