Bi-Hamiltonian structures of KdV type, cyclic Frobenius algebrae and Monge metrics

We study algebraic and projective geometric properties of Hamiltonian trios determined by a constant coefficient second-order operator and two first-order localizable operators of Ferapontov-Pavlov type. We show that first-order operators are determined by Monge metrics, and define a structure of cyclic Frobenius algebra. Examples include the AKNS system, a 2-component generalization of Camassa-Holm equation and the Kaup-Broer system. In dimension 2 the trio is completely determined by two conics of rank at least 2. We provide a partial classification in dimension 4.