This paper contains strong results concerning Kock-Zöberlein doctrines that satisfy a certain bicomma object condition (called admissible KZ-doctrines). In this context a notion of bifibration (which includes discrete fibrations and opfibrations) is introduced and studied. The main questions addressed are: the characterization of the Eilenberg-Moore algebras in terms of cocompleteness; the description of the Kleisli 2-category by means of its bifibrations; obtaining, in terms of bifibration, a “comprehensive” factorization of 1-cells (and 2-cells); the investigation of the case when the KZ-doctrine is stable under change of base (a characterization of the algebras as linear objects and a classification of discrete fibrations are obtained). Known facts about the symmetric monad are revisited and new results for complete spreads in topos theory are obtained. The main example is the symmetric monad in toposes, but also other KZ-doctrines are considered: the lower power locale, the lower bagdomain, the lifting monad in a posetal 2-category.