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We study algebraic and projective geometric properties of Hamiltonian triosdetermined by a constant coefficient second-order operator and two first-orderlocalizable operators of Ferapontov type. We show that first-order operatorsare determined by Monge metrics, and define a structure of cyclic Frobeniusalgebra. Examples include the AKNS system, a $2$-component generalization ofCamassa-Holm equation and the Kaup--Broer system. In dimension $2$ the trio iscompletely determined by two conics of rank at least $2$. We provide a partialclassification in dimension $4$.

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