In a recent research on degenerate points of steady axisymmetric gravity
flows with general vorticity, it has been shown that the possible asymptotics
near any stagnation point must be the "Stokes corner", the "horizontal cusp",
or the "horizontal flatness" (Theorem 1.1, Du, Huang, Pu, Commun. Math. Phys.,
400, 2137-2179, 2023). In this paper, we focus on the horizontally flat
singularity and show that it is not possible, and therefore the "Stokes corner"
and the "cusp" are the only possible asymptotics at the stagnation points. The
basic idea of our proof relies on a perturbation of the frequency formula for
the two-dimensional problem (Varvaruca, Weiss, Acta Math., 206, 363-403, 2011).
Our analysis also suggests that, for steady axisymmetric rotational gravity
flows, the singular asymptotic profiles at stagnation points are similar to the
scenario observed in two-dimensional waves with vorticity (Varvaruca, Weiss,
Ann. I. H. Poincare-AN, 29, 861-885, 2012)