  We extend the results of previous work on the vortex order parameter in
systems similar to the Ginzburg-Landau description of superfluid $^3$He in the
bulk B phase. Specifically, we consider vortices preserving an axial $U(1)$
symmetry. We determine the conditions required by the $\beta_i$ parameters to
allow for an energetically favorable development of the off-diagonal
antisymmetric and symmetric-traceless elements satisfying the axial symmetry
from the trace-only ansatz of the order parameter. The number and type of
gapless moduli appearing on the classical low energy theory of axial vortices
is determined. The time-dependent part of the Ginzburg-Landau free energy is
then considered to determine the number of quantized modes emerging from the
gapless moduli fields.
