  The analytic treatment of the non-inertial rotational diffusion equation,
i.e., of the Smoluchowski's one (SE), in a symmetric genuinely double-well
Maier-Saupe uniaxial potential of mean torque is considered. Such potential may
find applications to reorientations of the fragments of structure in polymers
and proteins. We obtain the exact solution of SE via the confluent Heun's
function. The solution is uniformly valid for any barrier height. We apply the
obtained solution to the calculation of the mean first passage time and the
longitudinal correlation time and obtain their precise dependence on the
barrier height. In the intermediate to high barrier (low temperature) region
the results of our approach are in full agreement with those of the approach
developed by Coffey, Kalmykov, D\'ejardin and their coauthors. In the low
barrier (high temperature) region our results noticeably distinguish from the
predictions of the literature formula and give appreciably greater values for
the transition rates from the potential well. The reason is that the above
mentioned formula is obtained in the stationary limit. We conclude that for
very small barrier heights the transient dynamics plays a crucial role and has
to be taken into account explicitly. When this requirement is satisfied (as,
e.g, at the calculation of the longitudinal correlation time) we obtain
absolute identity of our results with the literature formula in the whole range
of barrier heights. The drawbacks of our approach are its applicability only to
the symmetric potential and its inability to yield an analytical expression for
the smallest non-vanishing eigenvalue.
