  The variational Hartree-Fock-Bogoliubov (HFB) mean-field theory is the
starting point of various (ab initio) many-body methods dedicated to superfluid
systems. While taking the zero-pairing limit of HFB equations constitutes a
text-book problem when the system is of closed-(sub)shell character, it is
typically, although wrongly, thought to be ill-defined whenever the naive
filling of single-particle levels corresponds to an open-shell system.
  The present work demonstrates that the zero-pairing limit of an HFB state is
mathematically well-defined, independently of the closed- or open-shell
character of the system in the limit. Still, the nature of the limit state
strongly depends on the underlying shell structure and on the associated naive
filling reached in the zero-pairing limit for the particle number A of
interest. All the analytical findings are confirmed and illustrated
numerically.
  While HFB theory has been intensively scrutinized formally and numerically
over the last decades, it still uncovers unknown and somewhat unexpected
features. From this general perspective, the present analysis demonstrates that
HFB theory does not reduce to Hartree-Fock theory even when the pairing field
is driven to zero in the HFB Hamiltonian matrix.
