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.
