Exactly solvable relativistic Hartree-Fock equation

We consider a model in one space dimension which simulates the helium atom, and show that the relativistic Hartree-Fock (HF) equation for the model can be solved analytically. When the HF solution is used as a variational wave function by varying some of the parameters in it, the expectation value of the Hamiltonian exhibits saddle-point behavior around the HF values of the parameters.