  We study structure of the gapless points (diabolical points) at zero magnetic
field ($H_z=0$) of single-spin models with spin anisotropies. Nontrivial
appearance of diabolical points at finite transverse field $H_x$ has been
studied from the view point of interference of the Berry phase, and related
phenomena have been experimentally found in the single molecular magnet Fe$_8$.
We study effects of the orthorhombic single-ion anisotropy $E(S_+^2+S_-^2)$ and
find a symmetry associated with the degeneracy, which provides a clear picture
of the global structure of energy level diagram including the excited states.
Moreover, we study effects of the higher order anisotropy $C(S_+^4+S_-^4)$, and
find that, in contrast to the semiclassical limit $(S\to\infty)$, location of a
pair annihilation of the diabolical point does not coincides with a point at
which a pair of diabolical points appears in nonzero $H_y$ space(bifurcation
points). Distance between the annihilation and bifurcation points vanishes when
$S\to\infty$, which restores the semiclassical result. We obtain a complete
structure of the diabolical points in the $(C,H_x)$ plane.
