  Polarization of a vortex state fermion has rich structure due to the
nontrivial momentum distribution of wave function. This larger freedom provides
an unique opportunity to prepare fermions in exotic polarized states, which do
not exist for plane-wave state fermions. Based on the so called spin-orbit
state which was studied both theoretically and experimentally, we put forward a
peculiar vortex muon whose polarization exhibits $Z_n$ symmetry and study its
decay. We investigate the azimuthal distribution of the emitted electrons and
find that it exhibits the same symmetry ($Z_n$) as the initial state.
