Measures on Hilbert space invariant with respect to Hamiltonian flows

We study Hamiltonian flows in a real separable Hilbert space equipped with a symplectic structure. We investigate measures on the Hilbert space invariant with respect to the flows of completely integrable Hamiltonian systems and this allow us to describe Hamiltonian flows in phase space by means of unitary groups in the space of functions square integrable with respect to the invariant measure. The introduced measures, invariant with respect to the flows of completely integrable Hamiltonian systems, are applied for studying model linear Hamiltonian systems admitting singularities as an unbounded increasing of a kinetic energy in a finite time. Owing to such approach, the solutions of the Hamilton equations having singularities can be described by means of the phase flow in the extended phase space and by the corresponding Koopman representation of the unitary group.