  We study the properties of classical and quantum stable structures in a 3D
parameter space corresponding to the dissipative kicked top. This is a model
system in quantum and classical chaos that gives a starting point for many body
examples. We are able to identify the influence of these structures in the
spectra and eigenstates of the corresponding (super)operators. This provides
with a complementary view with respect to the typical 2D parameter space
systems found in the literature. Many properties of the eigenstates, like its
localization behaviour can be generalized to this higher dimensional parameter
space and spherical phase space topology. Moreover we find a 3D phenomenon --
generalizable to more dimensions -- that we call the coalescence-separation of
(q)ISSs, whose main consequence is a marked enhancement of quantum
localization. This could be of relevance for systems which have attracted a lot
of attention very recently.
