  We consider the gapless modes along the vortex line of the fully gapped,
momentum independent paired states of three-dimensional Dirac fermions. For
this, we require the solution of fermion zero modes of the corresponding
two-dimensional problem in the presence of a point vortex, in the plane
perpendicular to the vortex line. Based on the spectral symmetry requirement
for the existence of the zero mode, we identify the appropriate generalized
Jackiw-Rossi Hamiltonians for different paired states. A four-dimensional
generalized Jackiw-Rossi Hamiltonian possesses spectral symmetry with respect
to an antiunitary operator, and gives rise to a single zero mode only for the
{\em odd vorticity}, which is formally described by a $Z_2$ index. In the
presence of generic perturbations such as chemical potential, Dirac mass, and
Zeeman couplings, the associated two-dimensional problem for the odd parity
topological superconducting state maps onto {\em two} copies of generalized
Jackiw-Rossi Hamiltonian, and consequently an odd vortex binds two Majorana
fermions. In contrast, there are no zero energy states for the topologically
trivial $s$-wave superconductor in the presence of any chiral symmetry breaking
perturbation in the particle-hole channel, such as regular Dirac mass. We show
that the number of one-dimensional dispersive modes along the vortex line is
also determined by the index of the associated two-dimensional problem. For an
axial superfluid state in the presence of various perturbations, we discuss the
consequences of the $Z_2$ index on the anomaly equations.
