  Exact analytic solution for the probability distribution function of the
non-inertial rotational diffusion equation, i.e., of the Smoluchowski one, in a
symmetric Maier-Saupe uniaxial potential of mean torque is obtained via the
confluent Heun's function. Both the ordinary Maier-Saupe potential and the
double-well one with variable barrier width are considered. Thus, the present
article substantially extends the scope of the potentials amenable to the
treatment by reducing Smoluchowski equation to the confluent Heun's one. The
solution is uniformly valid for any barrier height. We use it for the
calculation of the mean first passage time. Also the higher eigenvalues for the
relaxation decay modes in the case of ordinary Maier-Saupe potential are
calculated. The results obtained are in full agreement with those of the
approach developed by Coffey, Kalmykov, D\'ejardin and their coauthors in the
whole range of barrier heights.
