  Recently F. Huang [Commun. Theor. Phys. V.42 (2004) 903] and X. Tang and P.K.
Shukla [Commun. Theor. Phys. V.49 (2008) 229] investigated symmetry properties
of the barotropic potential vorticity equation without forcing and dissipation
on the beta-plane. This equation is governed by two dimensionless parameters,
$F$ and $\beta$, representing the ratio of the characteristic length scale to
the Rossby radius of deformation and the variation of earth' angular rotation,
respectively. In the present paper it is shown that in the case $F\ne 0$ there
exists a well-defined point transformation to set $\beta = 0$. The
classification of one- and two-dimensional Lie subalgebras of the Lie symmetry
algebra of the potential vorticity equation is given for the parameter
combination $F\ne 0$ and $\beta = 0$. Based upon this classification, distinct
classes of group-invariant solutions is obtained and extended to the case
$\beta \ne 0$.
