A trivial result

This explains why normal mode analysis always gives doubly-degenerate results at the bottom of the wells. Freshman level proof, but the results are pretty unexpected.

A possible way to investigate the effects of symmetry breaking

With the five-fold symmetry about z-axis intact, the Poincaré surface of section of (φ=0, p_φ>0) will look identical to (φ=2nπ/5, p_φ>0), n=1, 2, 3, 4, ... but won't if the symmetry is broken. What happens if there is a strong localization? One may guess the density on the “right” surface of section is going to be much denser, but again the measure of the section points is always zero; one needs to think about a way to go about that.

Another issue: on generating initial conditions evenly on the surface of identical energy: if we first pick up r then v, the number of v's given r should be proportional to E⁰ in 2d and E^1/2 in 3d.