By solving analytically the various types of Lane-Emden equations with rotation, we have discovered two new coupled fundamental properties of rotating, self-gravitating, gaseous discs in equilibrium: isothermal discs must, on average, exhibit strict power-law density profiles in radius x on their equatorial planes of the form
, where
A and
k-1 are the integration constants; and “flat” rotation curves precisely such as those observed in spiral galaxy discs. Polytropic discs must, on average, exhibit strict density profiles of the form
, where
n is the polytropic index; and “flat” rotation curves described by square roots of upper incomplete gamma functions. By “on average”, we mean that, irrespective of the chosen boundary conditions, the actual profiles must oscillate around and remain close to the strict mean profiles of the analytic singular equilibrium solutions. We call such singular solutions the “intrinsic” solutions of the differential equations because they are demanded by the second-order equations themselves with no regard to the Cauchy problem. The results are directly applicable to gaseous galaxy discs that have long been known to be isothermal and to protoplanetary discs during the extended isothermal and adiabatic phases of their evolution. In galactic gas dynamics, they have the potential to resolve the dark matter—modified gravity controversy in a sweeping manner, as they render both of these hypotheses unnecessary. In protoplanetary disc research, they provide observers with a powerful new probing tool, as they predict a clear and simple connection between the radial density profiles and the rotation curves of self-gravitating discs in their very early (pre-Class 0) phases of evolution.