Mestel disk potential-density pair ($\Phi =v_0^2\log [(r+|z|)/R_{\ast }]$)

The reflection-symmetric Newtonian gravitational potential $\Phi =v_0^2\log [(\sqrt{R^2+z^2}+|z|)/R_{\ast }]$ is harmonic off the plane. Its normal-derivative jump is $2v_0^2/R$, so the Poisson equation for Newtonian gravity gives surface density of a disk $S=v_0^2/(2\pi GR)$ and volume mass density $\rho =S\delta (z)$. The midplane circular speed is constant. The scale-free model has infinite total mass and does not use a finite-mass boundary condition at infinity.

The midplane galaxy rotation curve fixes the radial derivative of the Newtonian gravitational potential, not every exterior boundary condition. Adding $a|z|$ leaves that radial derivative unchanged but adds a uniform sheet of surface density of a disk $a/(2\pi G)$. The pure scale-free Mestel disk excludes such extra sources.