  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 disks in equilibrium: Isothermal disks
must, on average, exhibit strict power-law density profiles in radius $x$ on
their equatorial planes of the form $A x^{k-1}$, where $A$ and $k-1$ are the
integration constants, and "flat" rotation curves precisely such as those
observed in spiral galaxy disks. Polytropic disks must, on average, exhibit
strict density profiles of the form $\left[\ln(A x^k)\right]^n$, 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 disks that have long been
known to be isothermal and to protoplanetary disks 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 disk research, they provide observers with
powerful new probing tool, as they predict a clear and simple connection
between the radial density profiles and the rotation curves of self-gravitating
disks in their very early (pre-Class 0 and perhaps the youngest Class Young
Stellar Objects) phases of evolution.
