Projective-geometric aspects of bi-Hamiltonian structures of KdV type

We introduce the problem of classification of bi-Hamiltonian structures of KdV type under projective reciprocal transformations. This problem leads naturally to studying the compatibility of a first order localizable homogeneous Hamiltonian operator with a higher order homogeneous Hamiltonian operator. We study the simplest third-order case where the orbit contains a constant operator. Computations with weakly non local Hamiltonian operators have been made by techniques developed in a previous paper.