Scale-free models of galaxies. III: A survey of the oblate E6 solution set

Numerous solutions of the collisionless Boltzmann and Poisson equations, having the same flattened axisymmetric scale-free mass distribution, are surveyed using Schwarzschild's self-consistent field method. A great variety of the self-similar models of very flat elliptical galaxies or invisible spiral halos, which these solutions constitute, are found, because orbits remaining close to the plane (due to large angular momenta) may be replaced by orbits with little angular momenta which are still well confined by the third integral. All solutions have velocity ellipsoids approximately aligned with the cylindrical coordinate axes, since the local dynamics is dominated by orbits near peri.