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An extended E⊗e Jahn-Teller Hamiltonian is presented for the case where the (slow) nuclear motion extends far from the symmetry point and may be described approximately as motion on a sphere. Rather than the traditional power series expansion in the displacement from the C 3v symmetry point, an expansion in the spherical harmonics is employed. Application is made to the vibrational Jahn-Teller effect in CH 3 XH, with X = S, O, where the equilibrium CXH angles are 83° and 72°, respectively. In addition to the symmetry-required conical intersection (CI) at the C 3v symmetry point, ab initio calculations reveal sets of six symmetry-allowed vibrational CIs in each molecule. The CIs for each molecule are arranged differently in the large-amplitude space, and that difference is reflected in the infrared spectra. The CIs in CH 3 SH are found in both eclipsed and staggered geometries, whereas those for CH 3 OH are found only in the eclipsed geometry near the torsional saddle point. This difference between the two molecules is reflected in the respective high-resolution spectra in the CH stretch fundamental region.