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 $\omega_{\lambda}=\omega_2+\lambda \omega_1=u\delta’(x-y)+\f{1}{2}u_x\delta(x-y)+\lambda\delta’(x-y)$. Deformations are generated by a sequence of vector fields $\{X_2,X_4,\mathrm{.}\mathrm{.}\mathrm{.}\}$, 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 ${\omega }_1$ and this tangency condition is equivalent to the exactness of the pencil ${\omega }_{\lambda }$. Moreover, extending the results of [17], we construct the non trivial deformations of the Poisson pencil ${\omega }_{\lambda }$, 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.