  We investigate the parameter dynamics of eigenvalues of Hamiltonians ('level
dynamics') defined on symmetric spaces relevant for condensed matter and
particle physics. In particular we: 1) identify appropriate reduced manifold on
which the motion takes place, 2) identify the correct Poisson structure
ensuring the Hamiltonian character of the reduced dynamics, 3) determine the
canonical measure on the reduced space, 4) calculate the resulting eigenvalue
density.
