  In this paper, we study axially symmetric solutions with $B=2-5$ in the
chiral quark soliton model.In the background of axially symmetric chiral
fields, the quark eigenstates and profile functions of the chiral fields are
computed self-consistently. The resultant quark bound spectrum are doubly
degenerate due to the symmetry of the chiral field. Upon quantization, various
observable spectra of the chiral solitons are obtained. Taking account of the
Finkelstein-Rubinstein constraints, we show that our results exactly coincide
with the physical observations for B=2 and 4 while B=3 and 5 do not.
