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 $N\le 4$. We finally comment how to extend this result for $N=6$.