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On bi-Hamiltonian deformations of exact pencils of hydrodynamic type

In this paper we are interested in non trivial bi-Hamiltonian deformations of the Poisson pencil $ω_λ=ω_2+λω_1=uδ'(x-y)+\f{1}{2}u_xδ(x-y)+λδ'(x-y)$. Deformations are generated by a sequence of vector fields $\{X_2, X_4,...\}$, where each $X_{2k}$ is homogenous of degree $2k$ with respect to a grading induced by rescaling. Constructing recursively the vector fields $X_{2k}$ 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 $X_{2k}$ to the symplectic leaves of $ω_1$ and this tangency condition is equivalent to the exactness of the pencil $ω_λ$. Moreover, extending the results of [17], we construct the non trivial deformations of the Poisson pencil $ω_λ$, 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.