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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) modelon the $GL(2,\mathbb{R})$ Lie group in twelve inequivalent families. Mostimportantly, it is shown that each of these models can be obtained from aPoisson-Lie T-dual $\sigma$-model in the presence of the spectator fields whenthe dual Lie group is considered to be Abelian, i.e. all deformed models havePoisson-Lie symmetry just as undeformed WZW model on the $GL(2,\mathbb{R})$. Inthis way, all deformed models are specified via spectator-dependent backgroundmatrices. For one case, the dual background is clearly found.

Yang-Baxter deformations of the $GL(2,\mathbb{R})$ WZW model and non-Abelian T-duality