Applying the <Poisson equation for Newtonian gravity> to a <flattened power-law gravitational potential> yields
$$
\rho=-\frac{\beta\Phi_0R_0^\beta}{4\pi G}\frac{(\lambda^2-\beta)R^2+\lambda^2[2-(\beta+1)\lambda^2]z^2}{(R^2+\lambda^2z^2)^{(\beta+4)/2}}.
$$
For $0<\beta<1$, nonnegative <mass density> everywhere outside the origin requires and is ensured by $\beta\leq\lambda^2\leq2/(\beta+1)$. The central density singularity is locally integrable and carries no point mass, although the total mass diverges at large radius in this scale-free model.


 Density positivity for a flattened power-law potential

ID: density-positivity-for-a-flattened-power-law-potential