The Lame polynomials naturally arise when separating variables in Laplace's
equation in elliptic coordinates. The products of these polynomials form a
class of spherical harmonics, which are joint eigenfunctions of a quantum
completely integrable (QCI) system of commuting, second-order differential
operators P-0 = Delta, P-1,...,PN-1 acting on C-infinity(S-N). These operat
ors naturally depend on parameters and thus constitute an ensemble, In this
paper, we compute the limiting level-spacings distributions for the zeroes
of the Lame polynomials in various thermodynamic, asymptotic regimes. We g
ive results both in the mean and pointwise, for an asymptotically full set
of values of the parameters.