In [5] I had established a connection between adiabatic invariants and the Gelfand--Lidskii signature of a symplectic matrix. The paper deals with linear Hamiltonian systems whose Hamiltonians have both  periodic and slow time--dependence: $\dot z = JH(t, \epsilon t)z$. One  question was: are there adiabatic invariants, and how many? The answer, in terms of the Gelfand--Lidskii signature, is this: if the periodic system $\dot z= JH(t, \tau)z$ (with $\tau$ frozen) is strongly stable for all $\tau$ then the number of adiabatic invariants of the system is not less than the number of clusters of the same sign in the Gelfand--Lidskii signature of the Floquet matrix.