Summary: Applying the classical Lie symmetry approach to the barotropic and quasi-geostrophic potential vorticity equation without forcing and dissipation on a \(\beta\)-plane channel, we find a new symmetry, which is not presented in a previous work [F. Huang, Commun. Theor. Phys. 42, No. 6, 903–908 (2004; Zbl 1167.76358)]. A general finite transformation group is obtained based on the full Lie point symmetry. Furthermore, two new types of similarity reduction solutions are obtained.