  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 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 $4$.
