  The possible stable singular vortex (SV) and half-quantum vortex (HQV) of the
superfluid $^3$He-A phase confined in restricted geometries are investigated.
The associated low-energy excitations are calculated in connection with the
possible existence of Majorana zero modes obeying non-Abelian statistics. The
energetics between those vortices is carefully examined using the standard
Ginzburg-Landau (GL) functional with a strong-coupling correction. The Fermi
liquid effect, which is not included in the GL functional, is considered
approximately within the London approach. This allows us to determine the
stability regions in pressure, temperature, and applied field for SV and HQV.
The existence of the Majorana zero mode and its statistics, either Abelian or
non-Abelian under braiding of SVs, is studied by solving the Bogoliubov-de
Gennes equation for spinful chiral p-wave superfluids at sufficiently low
temperatures. We determined several conditions controllable external parameters
for realizing the non-Abelian statistics of Majorana zero modes e.g., pressure,
field direction, and strength.
