Generic symmetries of the Laurent extension of quantum plane

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