  We demonstrate how conservation of longitudinal magnetization can have
pronounced effects on both stability and structure of vortices in the atomic
spin-1 Bose-Einstein condensate by providing a systematic characterization of
nonsingular and singular vortex states. Constructing spinor wave functions for
vortex states that continuously connect ferromagnetic and polar phases we
systematically derive analytic models for nonrotating cores of different
singular vortices and for composite defect states with distinct small- and
large-distance topology. We explain how the conservation law provides a
stabilizing mechanism when the coreless vortex imprinted on the condensate
relaxes in the polar regime of interatomic interactions. The resulting
structure forms a composite defect: the inner ferromagnetic coreless vortex
deforms toward an outer singly quantized polar vortex. We also numerically show
how other even more complex hierarchies of vortex core topologies may be
stabilized. Moreover, we analyze the structure of the coreless vortex also in a
ferromagnetic condensate, and show how reducing magnetization leads to a
displacement of the vortex from the trap center and eventually to the
deformation and splitting of its core where a singular vortex becomes a
lower-energy state. For the case of singular vortices, we find that the
stability and the core structure are notably less influenced by the
conservation of magnetization.
