  Using both numerical and analytical tools we study various features of
static, spherically symmetric solutions of the Einstein-Vlasov system. In
particular, we investigate the possible shapes of their mass-energy density and
find that they can be multi-peaked, we give numerical evidence and a partial
proof for the conjecture that the Buchdahl inequality $\sup_{r > 0} 2 m(r)/r <
8/9$, $m(r)$ the quasi-local mass, holds for all such steady states--both
isotropic {\em and} anisotropic--, and we give numerical evidence and a partial
proof for the conjecture that for any given microscopic equation of state--both
isotropic {\em and} anisotropic--the resulting one-parameter family of static
solutions generates a spiral in the radius-mass diagram.
