from
those in the vortex solution, 
In local rotated coordinate
frames, 
, where the axis 
coincides with the 
-vector's direction in the
unperturbed vortex, and
the variables 
are the projections
of the 
-vector on the local axes 
and 
: 
.
It is easy to get the coupled set of two partial differential
equations for 
and 
:
where
, and the ``potentials'' are:\
.
Using the ansatz like that in [8],
where 
and
a,b are arbitrary constants, one obtains a coupled set of
ODEs for 
and 
.
Here 
...; k and m are the full set of
quantum numbers denoting the eigenvalues 
The presence of the combination of exponentials 
with arbitrary coefficients is due to the degeneracy of
the modes with m=+|m| and m=-|m|.
One can see that the coupling of 
and 
in Eq.
(6) comes from the term with 
only. This means that (i) the
coupling vanishes exponentially at 
(
);
(ii) there is no coupling for m=0 modes, in contrast to
the FM case [8],[12].
