We all know that in 1 dimensional classical mechanics, a mass in the presence of a potential 1/2mw²x² will perform an harmonic motion with period 2Π/w. The period is independent of the total energy.
Are there other potentials, where the period of the bound states does not depend of the energy?

Yes.

Did you find it in the same book that I found it?

No, I had never heard it before, very interesting though. I came up with a few counterexamples after I thought about it for a minute. Where did you read about it?

Woops, my answer was supposed to be in white above. Some of the simple ones I thought of were a half harmonic oscillator half infinite wall and an asymmetric harmonic oscillator (two different frequencies for x < 0 and x > 0).