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We demonstrate that linear degeneracy is a necessary condition forquasilinear systems of Jordan block type to possess first-order Hamiltonianstructures. Multi-Hamiltonian formulation of linearly degenerate systemsgoverning delta-functional reductions of the kinetic equation for dense solitongas is established (for KdV, sinh-Gordon, hard-rod, Lieb-Liniger, DNLS, andseparable cases).

Hamiltonian systems of Jordan block type: delta-functional reductions of the kinetic equation for soliton gas