  We show, by means of numerical and analytical methods, that media with a
repulsive nonlinearity which grows from the center to the periphery support a
remarkable variety of previously unknown complex stationary and dynamical
three-dimensional solitary-wave states. Peanut-shaped modulation profiles give
rise to vertically symmetric and antisymmetric vortex states, and novel
stationary hybrid states, built of top and bottom vortices with opposite
topological charges, as well as robust dynamical hybrids, which feature stable
precession of a vortex on top of a zero-vorticity base. The analysis reveals
stability regions for symmetric, antisymmetric, and hybrid states. In addition,
bead-shaped modulation profiles give rise to the first example of exact
analytical solutions for stable three-dimensional vortex solitons. The
predicted states may be realized in media with a controllable cubic
nonlinearity, such as Bose-Einstein condensates.
