  Toward a unified algebraic theory for mean-field Hamiltonian (MFH) describing
paired- and unpaired-mode effects, in this paper, we propose a generalized HB
(GHB) MFH in terms of the SO(2N+1)Lie algebra of fermion pair and
creation-annihilation operators. We diagonalize the GHB-MFH and throughout the
diagonalization of which, we can first obtain the unpaired mode amplitudes
which are given by the SCF parameters appeared in the HBT together with the
additional SCF parameter in the GHB-MFH and by the parameter specifying the
property of the SO(2N +1) group. Consequently, it turns out that the magnitudes
of these amplitudes are governed by such parameters. Thus, it becomes possible
to make clear a new aspect of such the results. We construct the Killing
potential in the coset space SO(2N)/U(N) on the Kaehler symmetric space which
is equivalent with the generalized density matrix (GDM). We show another
approach to the fermion MFH based on such a GDM. We derive an SO(2N +1) GHB MF
operator and a modified HB eigenvalue equation. We discuss on the MF theory
related to the algebraic MF theory based on the GDM and the coadjoint orbit
leading to the non-degenerate symplectic form.
