  We use the vorticity transportation equation as the start point--with the
help of stream function for two-dimensional planar incompressible flows--to
obtain exact solutions that characterize evolution and dynamics of the flows.
These traveling wave solutions, including both continuous and solitary waves,
also recover the results existing in the theory of potential flows. We further
analyze the flow patterns regarding symmetry solutions generated by Lie
transformation groups. Some of the symmetry solutions have not been discussed
in the current literature. They could display flow patterns very different from
that of the original exact solutions, suggesting the symmetry solutions are not
trivial. Particularly, the symmetry solutions to Lame-Oseen vortex and doublet
exhibit more complicated dynamics.
