  By calculating inequivalent classical r-matrices for the $gl(2,\mathbb{R})$
Lie algebra as solutions of (modified) classical Yang-Baxter equation
((m)CYBE)), we classify the YB deformations of Wess-Zumino-Witten (WZW) model
on the $GL(2,\mathbb{R})$ Lie group in twelve inequivalent families. Most
importantly, it is shown that each of these models can be obtained from a
Poisson-Lie T-dual $\sigma$-model in the presence of the spectator fields when
the dual Lie group is considered to be Abelian, i.e. all deformed models have
Poisson-Lie symmetry just as undeformed WZW model on the $GL(2,\mathbb{R})$. In
this way, all deformed models are specified via spectator-dependent background
matrices. For one case, the dual background is clearly found.
