  Let $v$ be a solution of the axially symmetric Navier-Stokes equation. We
determine the structure of certain (possible) maximal singularity of $v$ in the
following sense. Let $(x_0, t_0)$ be a point where the flow speed $Q_0 =
|v(x_0, t_0)|$ is comparable with the maximum flow speed at and before time
$t_0$. We show after a space-time scaling with the factor $Q_0$ and the center
$(x_0, t_0)$, the solution is arbitrarily close in $C^{2, 1, \alpha}_{{\rm
local}}$ norm to a nonzero constant vector in a fixed parabolic cube, provided
that $r_0 Q_0$ is sufficiently large. Here $r_0$ is the distance from $x_0$ to
the $z$ axis. Similar results are also shown to be valid if $|r_0v(x_0, t_0)|$
is comparable with the maximum of $|rv(x, t)|$ at and before time $t_0$.
