The essential coexistence phenomenon in Hamiltonian dynamics

We construct an example of a Hamiltonian flow on a four-dimensional smooth manifold which after being restricted to an energy surface demonstrates essential coexistence of regular and chaotic dynamics, that is, there is an open and dense-invariant subset such that the restriction has non-zero Lyapunov exponents in all directions (except for the direction of the flow) and is a Bernoulli flow while, on the boundary, which has positive volume, all Lyapunov exponents of the system are zero.