  The superfluid $^3$He formed by spin-triplet $p$-wave Cooper pairs is a
typical topological superfluid. In the superfluid $^3$He B-phase, several kinds
of vortices classified by spatial symmetries $P_1$, $P_2$, and $P_3$ are
produced, where $P_1$ is inversion symmetry, $P_2$ is magnetic reflection
symmetry, and $P_3$ is magnetic $\pi$-rotation symmetry. We have calculated the
vortex bound states by the Bogoliubov-de Gennes theory and the quasiclassical
Eilenberger theory, and also clarified symmetry protection of the low energy
excitations by the spatial symmetries. On the symmetry protection, $P_3$
symmetry plays a key role which gives two-fold degenerate Majorana zero modes.
Then, the bound states in the most symmetric $o$ vortex with $P_1$, $P_2$, and
$P_3$ symmetries and in $w$ vortex with $P_3$ symmetry have the symmetry
protected degenerate Majorana zero modes. On the other hand, zero energy modes
in $v$ vortex, which is believed to be realized in the actual B-phase, are not
protected, and in consequence become gapped by breaking axial symmetry. The
excitation gap may have been observed as the variation of critical velocity. We
have also suggested an experimental setup to create $o$ vortex with Majorana
zero modes by a confinement and a magnetic field.
