  We prove local blow-up criterion for smooth axisymmetric solutions to the 3D
incompressible Euler equation. If the vorticity satisfies $ \intl_{0}^{t_*}
(t_*-t) \| \omega (t)\|_{ L^\infty(B(x_{ \ast}, R_0))} dt <+\infty$ for a ball
$B(x_{ \ast}, R_0)$ away from the axis of symmetry, then there exists no
singularity at $t=t_*$ in the torus $T(x_*, R)$ generated by rotation of the
ball $B(x_{ \ast}, R_0)$ around the axis. This implies that possible
singularity at $t=t_*$ in the torus $T(x_*, R)$ is excluded if the vorticity
satisfies the blow-up rate $ \|\o (t)\|_{L^\infty (T(x_*, R))}=
O\left(\frac{1}{(t_*-t)^\gamma}\right)$ as $t\to t_*$, where $\gamma <2$ and
the torus $T(x_*, R)$ does not touch the axis.
