  It was shown in \cite{FPV} that the classification of $n$-component systems
of conservation laws possessing a third-order Hamiltonian structure reduces to
the following algebraic problem: classify $n$-planes $H$ in $\wedge^2(V_{n+2})$
such that the induced map $Sym^2H\longrightarrow \wedge^4V_{n+2}$ has
1-dimensional kernel generated by a non-degenerate quadratic form on $H^*$.
This problem is trivial for $n=2, 3$ and apparently wild for $n\geq 5$. In this
paper we address the most interesting borderline case $n=4$. We prove that the
variety $\mathcal{V}$ parametrizing those 4-planes $H$ is an irreducible
38-dimensional $PGL(V_6)$-invariant subvariety of the Grassmannian $G(4,
\wedge^2V_6)$. With every $H\in\mathcal{V}$ we associate a {\it characteristic}
cubic surface $S_H\subset \mathbb{P} H$, the locus of rank 4 two-forms in $ H$.
We demonstrate that the induced characteristic map $\sigma: \mathcal{V} /
PGL(V_6) \dashrightarrow \mathcal{M}_c,$ where $\mathcal{M}_c$ denotes the
moduli space of cubic surfaces in $\mathbb{P}^3$, is dominant, hence
generically finite. A complete classification of 4-planes $H\in\mathcal{V}$
with the reducible characteristic surface $S_H$ is given.
