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This work finishes a classification of $U_q(\mathfrak{sl}_2)$-symmetries onthe Laurent extension $\mathbb{C}_q\big[x^{\pm 1},y^{\pm 1}\big]$ of thequantum plane. After reproducing the partial results of a previous paper of theauthor related to symmetries with non-trivial action of the Cartan generator(s)of $U_q(\mathfrak{sl}_2)$ and the generic symmetries, a complete collection ofnon-generic symmetries is presented. Together, these collections constitute acomplete list of $U_q(\mathfrak{sl}_2)$-symmetries on $\mathbb{C}_q\big[x^{\pm1},y^{\pm 1}\big]$.

The Laurent Extension of Quantum Plane: a Complete List of $U_q(\mathfrak{sl}_2)$-Symmetries