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

A new description of motion of the Fermionic SO(2N+2) top in the classical limit under the quasi-anticommutation relation approximation