Line geometry of pairs of second-order Hamiltonian operators and quasilinear systems

We show that a pair formed by a second-order homogeneous Hamiltonian structures in N components and the associated system of conservation laws is in bijective correspondence with an alternating three-form on a N+2 vector space. We use this result to characterise these pairs up to N=4. We also show that the three-form provides N+2 linear equations in the Pl\"ucker coordinates which define the associated line congruence.