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

3-D off-center Li+ rotors in alkali halides: solving for the eigenvalue problem