In this paper, we introduce a generalized Hartree-Bogoliubov (GHB) mean-field Hamiltonian (MFH) in terms of the SO(2N+1) Lie algebra of fermion operators and diagonalize the GHB MFH. Throughout the diagonalization, we could first obtain solutions for unpaired mode amplitudes using the already solved solutions for the fermion HB equation for paired mode amplitudes and we make clear a new aspect of such the solutions. We also study the Kaehler symmetric space SO(2N)/U(N). We construct a Killing potential in the coset space SO(2N)/U(N) which is equivalent with the generalized density matrix (GDM). We further propose 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. Further we discuss on the MF theory related to the algebraic MF theory based on the GDM and the coadjoint orbit which leads to the non-degenerate symplectic form.