  We study the behavior of Hartree-Fock (HF) solutions in the vicinity of
conical intersections. These are here understood as regions of a molecular
potential energy surface characterized by degenerate or nearly-degenerate
eigenfunctions with identical quantum numbers (point group, spin, and electron
number). Accidental degeneracies between states with different quantum numbers
are known to induce symmetry breaking in HF. The most common closed-shell
restricted HF instability is related to singlet-triplet spin degeneracies that
lead to collinear unrestricted HF (UHF) solutions. Adding geometric frustration
to the mix usually results in noncollinear generalized HF (GHF) solutions,
identified by orbitals that are linear combinations of up and down spins. Near
conical intersections, we observe the appearance of coplanar GHF solutions that
break all symmetries, including complex conjugation and time-reversal, which do
not carry good quantum numbers. We discuss several prototypical examples taken
from the conical intersection literature. Additionally, we utilize a recently
introduced a magnetization diagnostic to characterize these solutions, as well
as a solution of a Jahn-Teller active geometry of H$_8^{+2}$.
