  In this paper we study the distribution of level crossings for the spectra of
linear families A+lambda B, where A and B are square matrices independently
chosen from some given Gaussian ensemble and lambda is a complex-valued
parameter. We formulate a number of theoretical and numerical results for the
classical Gaussian ensembles and some generalisations. Besides, we present
intriguing numerical information about the distribution of monodromy in case of
linear families for the classical Gaussian ensembles of 3 * 3 matrices.
