  It is well known that the simple criterion proposed originally by Polyachenko
and Shukhman (1981) for the onset of the radial orbit instability, although
being generally a useful tool, faces significant exceptions both on the side of
mildly anisotropic systems (with some that can be proved to be unstable) and on
the side of strongly anisotropic models (with some that can be shown to be
stable). In this paper we address two issues: Are there processes of
collisionless collapse that can lead to equilibria of the exceptional type?
What is the intrinsic structural property that is responsible for the sometimes
noted exceptional stability behavior? To clarify these issues, we have
performed a series of simulations of collisionless collapse that start from
homogeneous, highly symmetrized, cold initial conditions and, because of such
special conditions, are characterized by very little mixing. For these runs,
the end-states can be associated with large values of the global pressure
anisotropy parameter up to 2K_r/K_T \approx 2.75. The highly anisotropic
equilibrium states thus constructed show no significant traces of radial
anisotropy in their central region, with a very sharp transition to a radially
anisotropic envelope occurring well inside the half-mass radius (around 0.2
r_M). To check whether the existence of such almost perfectly isotropic
"nucleus" might be responsible for the apparent suppression of the radial orbit
instability, we could not resort to equilibrium models with the above
characteristics and with analytically available distribution function; instead,
we studied and confirmed the stability of configurations with those
characteristics by initializing N-body approximate equilibria (with given
density and pressure anisotropy profiles) with the help of the Jeans equations.
