  General criteria to check the positivity of the distribution function
(phase-space consistency) of stellar systems of assigned density and anisotropy
profile are useful starting points in Jeans-based modeling. Here we
substantially extend previous results, and we present the inversion formula and
the analytical necessary and sufficient conditions for phase-space consistency
of the family of multi-component Cuddeford spherical systems: the distribution
function of each density component of these systems is defined as the sum of an
arbitrary number of Cuddeford distribution functions with arbitrary values of
the anisotropy radius, but identical angular momentum exponent. The radial
trend of anisotropy that can be realized by these models is therefore very
general. As a surprising by-product of our study, we found that the ``central
cusp-anisotropy theorem'' (a necessary condition for consistency relating the
values of the central density slope and of the anisotropy parameter) holds not
only at the center, but at all radii in consistent multi-component generalized
Cuddeford systems. This last result suggests that the so--called
mass--anisotropy degeneracy could be less severe than what is sometimes feared.

Cosmology and Nongalactic Astrophysics