We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. % Examples include equations of associativity of two-dimensional topological % field theory (WDVV equations). and various equations of Monge-Amp\`ere % 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 $\phi$.