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The operator in the Hamiltonian which dominates the fine-structure splitting patterns in spherical-top molecules of tetrahedral and octahedral symmetry is the irreducible tensor $$T = (70)^{1/2}T(40) + 5[T(44) + T(4-4)]$$ which can be expressed, for example, in the case of pure rotations as \begin{eqnarray*} -6(J^{2})^{2} + 10 (J^{4}{x} + J^{4}{y} + J^{4}{z}) + 2J^{2}\ = {3\over2} (J^{2})^{2} - 15J^{2}J^{2}{o} + {35 \over 2}J^{4} - 3J^{2} + {25 /over 2}J^{2}{o} +{ 5 \over 4} (J^{4}{+} + J^{4}_{-}) .\end{eqnarray*} The eigenvalues of this operator have been used to account for complicated high resolution infrared spectra of molecules like $CH4$ and $SF6$. We have extended the calculations of these eigenvalues to $J\le 100$, and have discovered regularities in the eigenvalue spectrum which persist for all values of J considered. Furthermore, other remarkable patterns appear at moderate values of J and become more pronounced as J increases. These phenomena will be illustrated with numerical examples.