  We explore the existence and behaviour of holomorphic restricted Hartree-Fock
(h-RHF) solutions for two-electron problems. Through algebraic geometry, the
exact number of solutions with $n$ basis functions is rigorously identified as
$\frac{1}{2}(3^n - 1)$, proving that states must exist for all molecular
geometries. A detailed study on the h-RHF states of HZ (STO-3G) then
demonstrates both the conservation of holomorphic solutions as geometry or
atomic charges are varied and the emergence of complex h-RHF solutions at
coalescence points. Using catastrophe theory, the nature of these coalescence
points is described, highlighting the influence of molecular symmetry. The
h-RHF states of HHeH$^{2+}$ and HHeH (STO-3G) are then compared, illustrating
the isomorphism between systems with two electrons and two electron holes.
Finally, we explore the h-RHF states of ethene (STO-3G) by considering the
$\pi$ electrons as a two-electron problem, and employ NOCI to identify a
crossing of the lowest energy singlet and triplet states at the perpendicular
geometry.
