  We have derived exact axisymmetric solutions of the two-dimensional
Lane-Emden equations with rotation. These solutions are intrinsically favored
by the differential equations regardless of any adopted boundary conditions and
the physical solutions of the Cauchy problem are bound to oscillate about and
remain close to these intrinsic solutions. The isothermal solutions are
described by power-law density profiles in the radial direction, whereas the
polytropic solutions are described by radial density profiles that are powers
of the zeroth-order Bessel function of the first kind. Both families of
solutions decay exponentially in the vertical direction and both result in
increasing or nearly flat radial rotation curves. The results are applicable to
gaseous spiral-galaxy disks that exhibit flat rotation curves and to the early
stages of protoplanetary disk formation before the central star is formed.
