  Rotation is ubiquitous in the Universe, and recent kinematic surveys have
shown that early type galaxies and globular clusters are no exception. Yet the
linear response of spheroidal rotating stellar systems has seldom been studied.
This paper takes a step in this direction by considering the behaviour of
spherically symmetric systems with differential rotation. Specifically, the
stability of several sequences of Plummer spheres is investigated, in which the
total angular momentum, as well as the degree and flavour of anisotropy in the
velocity space are varied. To that end, the response matrix method is
customised to spherical rotating equilibria. The shapes, pattern speeds and
growth rates of the systems' unstable modes are computed. Detailed comparisons
to appropriate N-body measurements are also presented. The marginal stability
boundary is charted in the parameter space of velocity anisotropy and rotation
rate. When rotation is introduced, two sequences of growing modes are
identified corresponding to radially and tangentially-biased anisotropic
spheres respectively. For radially anisotropic spheres, growing modes occur on
two intersecting surfaces (in the parameter space of anisotropy and rotation),
which correspond to fast and slow modes, depending on the net rotation rate.
Generalised, approximate stability criteria are finally presented.
