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

We demonstrate that a pair consisting of a second-order homogeneous Hamiltonian structure in N components and its associated system of conservation laws is in bijective correspondence with an alternating three-form on a N+2-dimensional vector space. Additionally, we show that the three-form offers N+2 linear equations in the Plücker coordinates that define the associated line congruence. We utilize these results to characterize systems of conservation laws with second-order structure for N≤4. We finally comment on how to extend this result for N=6.