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

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