  The coalescence and disappearance of Hartree--Fock (HF) solutions as the
molecular structure varies have been a common source of criticism for the
breakdown of the HF approximation to the potential energy surfaces. However,
recent developments in holomorphic HF theory show that this disappearing
behavior is only a manifestation of the way conventional HF equations prevent
solutions from being analytically continued, but it is unclear what factors
govern the existence and the locations of these disappearances. In this work,
we explore some of these factors from the perspective of spatial symmetry by
introducing a classification for symmetry constraints on electronic-structure
calculations. This forms a framework for us to systematically investigate
several analytic holomorphic HF solutions of a model
$\textrm{[H}_4\textrm{]}^{2+}$ system in STO-3G and demonstrate that, under
appropriate conditions, spatial symmetry imposes strict requirements on the
reality of certain solutions. The implications for self-consistent-field HF
search algorithms are then discussed. Throughout this article, the term reality
means the quality of a holomorphic HF solution having real molecular orbitals.
