  In this paper, we investigate the singularity near the degenerate points of
the steady axisymmetric flow with general vorticity of an inviscid
incompressible fluid acted on by gravity and with a free surface. We called the
points on the free boundary at which the gradient of the stream function
vanishes as the degenerate points. The main results in this paper give the
different classifications of the singularity near the degenerate points on the
free surface. More precisely, we obtained that at the stagnation points, the
possible profiles must be a Stokes corner, or a horizontal cusp, or a
horizontal flatness. At the degenerate points on the symmetric axis except the
origin, the wave profile must be a cusp. At the origin, the possible wave
profiles must be a Garabedian pointed bubble, or a horizontal cusp, or a
horizontal flatness.
