On bi-Hamiltonian deformations of exact pencils of hydrodynamic type

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In this paper we are interested in non trivial bi-Hamiltonian deformations of the Poisson pencil ωλ=ω₂+λω₁=uδ'(x-y)+12uₓδ(x-y)+λδ'(x-y). Deformations are generated by a sequence of vector fields {X₂, X₄,...}, where each X₂ₖ is homogenous of degree 2k with respect to a grading induced by rescaling. Constructing recursively the vector fields X₂ₖ one obtains two types of relations involving their unknown coefficients: one set of linear relations and an other one which involves quadratic relations. We prove that the set of linear relations has a geometric meaning: using Miura-quasitriviality the set of linear relations expresses the tangency of the vector fields X₂ₖ to the symplectic leaves of ω₁ and this tangency condition is equivalent to the exactness of the pencil ωλ. Moreover, extending the results of [17], we construct the non trivial deformations of the Poisson pencil ωλ, up to the eighth order in the deformation parameter, showing therefore that deformations are unobstructed and that both Poisson structures are polynomial in the derivatives of u up to that order.