  [Abridged] Recently we have found that a family of models of partially
relaxed, anisotropic stellar systems, inspired earlier by studies of incomplete
violent relaxation, exhibits some interesting thermodynamic properties. Here we
present a systematic investigation of its dynamical characteristics, in order
to establish the basis for a detailed comparison with simulations of
collisionless collapse, planned for a separate paper. For a full comparison
with the observations of elliptical galaxies, the models should be extended to
allow for the presence a sizable dark halo and of significant rotation. In the
spherical limit, the family is characterized by two dimensionless parameters,
i.e. \Psi, measuring the depth of the galaxy potential, and \nu, defining the
form of a third global quantity Q, which is argued to be approximately
conserved during collisionless collapse (in addition to the total energy and
the total number of stars).
