The authors continue the study of fine properties of spectra of random matrix ensembles of special types. These are characterized by the explicit form of the joint distribution of eigenvalues of random matrices; the cases considered are the Gaussian unitary ensemble and the symmetric Jacobi unitary ensemble. Basing on the form of the eigenvalue distribution, a nonlinear ordinary differential equations formalism is developed to obtain expressions for the probabilities for gaps in the eigenvalue spectrum. Relations with the Painlevé transcendents are discussed.