  The boson images of fermion SO(2N+1) Lie operators have been given together
with those of SO(2N+2) ones. The SO(2N+1) Lie operators are generators of
rotation in the (2N+1)-dimensional Euclidian space (N: number of
single-particle states of the fermions). The images of fermion
annihilation-creation operators must satisfy the canonical anti-commutation
relations, when they operate on a spinor subspace. In the regular
representation space we use a boson Hamiltonian with Lagrange multipliers to
select out the spinor subspace. Based on these facts, a new description of a
fermionic SO(2N+2) top is proposed. From the Heisenberg equations of motions
for the boson operators, we get the SO(2N+1) self-consistent field (SCF)
Hartree-Bogoliubov (HB) equation for the classical stationary motion of the
fermion top. Decomposing an SO(2N+1) matrix into matrices describing paired and
unpaired modes of fermions, we obtain a new form of the SO(2N+1) SCF equation
with respect to the paired-mode amplitudes. To demonstrate the effectiveness of
the new description based on the bosonization theory, the extended HB
eigenvalue equation is applied to a superconducting toy-model which consists of
a particle-hole plus BCS type interaction. It is solved to reach an interesting
and exciting solution which is not found in the traditional HB eigenvalue
equation, due to the unpaired-mode effects. To complete the new description,
the Lagrange multipliers must be determined in the classical limit. For this
aim a quasi anti-commutation-relation approximation is proposed. Only if a
certain relation between an SO(2N+1) parameter z and the N is satisfied,
unknown parameters k and l in the Lagrange multipliers can be determined
withuout any inconcistency.
