First-order Hamiltonian operators of hydrodynamic type were introduced by Drubrovin and Novikov in 1983. In 2D, they are generated by a pair of contravariant metrics $g$, $\tilde{ g}$ and a pair of differential-geometric objects $b$, $\tilde{ b}$. If the determinant of the pencil $g+\lambda \tilde{ g}$ vanishes for all $\lambda$, the operator is called degenerate. In this paper we provide a complete classification of degenerate two- and three-component Hamiltonian operators. Moreover, we study the integrability, by the method of hydrodynamic reductions, of 2+1 Hamiltonian systems arising from the structures we classified.
Categories: math-ph math.MP nlin.SI