  We demonstrate that linear degeneracy is a necessary condition for
quasilinear systems of Jordan block type to possess first-order Hamiltonian
structures. Multi-Hamiltonian formulation of linearly degenerate systems
governing delta-functional reductions of the kinetic equation for dense soliton
gas is established (for KdV, sinh-Gordon, hard-rod, Lieb-Liniger, DNLS, and
separable cases).
