We consider unitary invariant random matrix ensembles which obey spectral statistics different from the Wigner-Dyson, including unitary ensembles with slowly (∼ log 2 x) growing potentials and the finite-temperature fermi gas model. If the deformation parameters in these matrix ensembles are small, the asymptotically translational-invariant region in the spectral bulk is universally governed by a one-parameter generalization of the sine kernel. We provide an analytic expression for the distribution of the eigenvalue spacings of this universal asymptotic kernel, which is a hybrid of the Wigner-Dyson and the Poisson distributions, by determining the Fredholm determinant of the universal kernel in terms of a Painlevé VI transcendental function. [4]. A characteristic observable in such studies, used analytically or numerically to measure the deviation from integrability, is the probability E(s) of having no energy levels in an interval of width s, or the distribution of spacings between adjacent levels P (s) = E ′′ (s). These observables capture the behavior of local correlations of large number of energy levels, as the former consists of an infinite sum of integrals of regulated † spectral correlators,