# Systems of conservation laws with third-order Hamiltonian structures

## TL;DR

This paper classifies multi-component conservation law systems with third-order Hamiltonian structures using geometric and algebraic methods, linking their properties to projective geometry and skew-symmetric forms.

## Contribution

It provides a geometric classification of such systems by reducing the problem to projective line congruences and algebraic conditions on skew-symmetric forms.

## Key findings

- Classification reduces to projective geometry of line congruences.
- Algebraic reformulation involves skew-symmetric 2-forms satisfying specific conditions.
- Establishes a link between Hamiltonian structures and geometric configurations.

## Abstract

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 $\phi$.