The Hartree-Fock equation which is the Euler-Lagrange equation corresponding
    to the Hartree-Fock energy functional is used in many-electron problems. Since
    the Hartree-Fock equation is a system of nonlinear eigenvalue problems, the
    study of structures of sets of all solutions needs new methods different from
    that for the set of eigenfunctions of linear operators. In this paper we prove
    that the sets of all solutions to the Hartree-Fock equation associated with
    critical values of the Hartree-Fock energy functional less than the first
    energy threshold are unions of compact real-analytic subsets of a finite number
    of connected real-analytic manifolds. The result would also be a basis for the
    study of approximation methods to solve the equation.

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