Generic symmetries of the Laurent extension of quantum plane

A list of generic $U_q(\mathfrak{sl}_2)$-module algebra structures on the Laurent polynomial algebra over the quantum plane with uncountably many isomorphism classes is produced. Also, a complete list of these structures is presented in which the action of Cartan generator of $U_q(\mathfrak{sl}_2)$ is not reduced to multiplying $x$ and $y$ (the generators of quantum plane) by constants.