  We investigate $n$-component systems of conservation laws that possess
third-order Hamiltonian structures of differential-geometric type. The
classification of such systems is reduced to the projective classification of
linear congruences of lines in $\mathbb{P}^{n+2}$ satisfying additional
geometric constraints. Algebraically, the problem can be reformulated as
follows: for a vector space $W$ of dimension $n+2$, classify $n$-tuples of
skew-symmetric 2-forms $A^{\alpha} \in \Lambda^2(W)$ such that \[ \phi_{\beta
\gamma}A^{\beta}\wedge A^{\gamma}=0, \] for some non-degenerate symmetric
$\phi$.
