## Non-Gaussian energy level statistics for some integrable systems

The number of levels with energy less than [ital E] of an integrable quantum system with two degrees of freedom is equal to [lambda][ital E]+[ital sE][sup 1/4], where [lambda] is a constant and [ital s] a fluctuating quantity with a non-Gaussian distribution. The probability distribution of [ital s] decreases roughly like exp([minus][ital s][sup 4]) when [ital s] is large. The number of levels between [ital E] and [ital E]+[ital z] [radical][ital E] is equal to [lambda][ital z] [radical][ital E] +[ital rE][sup 1/4] where [ital r] is another fluctuating quantity. The distribution of [ital r] tends to a Gaussian distribution asmore »