  We are dealing with the long-standing problem of the eigenstates and
eigenvalues of 3-D rotators by off-center impurity ions in alkali halides. The
ion runs along the brim of a sombrero-like vibronic potential. The
quantum-mechanical motion depending on two angular coordinates, we consider the
motion along theta at averaged phi, and vice versa. We find the off-center ion
performing two rotations: one in (x,y) plane along the azimuth phi and another
one along the position angle theta, respectively, which are both described as
Mathieu's non-linear oscillators. The motion along z is complemented by an
anharmonic vibrational mode.
