  We modify the curvature condition for the existence of self-consistent
scale-free discs, introduced by Zhao, Carollo & de Zeeuw. We survey the
parameter space of the power-law discs, and show that the modified curvature
condition is in harmony with the results of Schwarzschild's numerical orbit
superposition method. We study the orbital structure of the power-law discs,
and find a correlation between the population of centrophobic banana orbits and
the non-self-consistency index. We apply the curvature condition to other
families of scale-free elongated discs and find that it rules out a large range
of power-law slopes and axis ratios. We generalize the condition, and apply it,
to three-dimensional scale-free axisymmetric galaxy models.

Astrophysics