  We consider the iterative solution of symmetric saddle point systems with a
rank-deficient leading block. We develop two preconditioners that, under
certain assumptions on the rank structure of the system, yield a preconditioned
matrix with a constant number of eigenvalues. We then derive some properties of
the inverse of a particular class of saddle point system and exploit these to
develop a third preconditioner, which remains ideal even when the earlier
assumptions on rank structure are relaxed.
