This is a situation which one encounters in for example a fermionic system like a closed-shell nucleus like oxygen-16 or a neutral noble gas like helium or neon with the same number of spin-up and spin-down orbitals and the same spatial single-particle functions. The example discussed above for neutral Beryllium where we fill the hydrogen-like states 1s and 2s , is yet another case.
For those of you familiar with Hartree-Fock theory, this is often referred to as restricted Hartree-Fock theory. Unrestricted Hartree-Fock theory represents then the more general case.