On bi-Hamiltonian deformations of exact pencils of hydrodynamic type

In this paper, we are interested in nontrivial bi-Hamiltonian deformations of the Poisson pencil omega(lambda) = omega(2) + lambda omega(1) = u delta' (x - y) + 1/2u(x) delta(x - y) + lambda delta'(x - y). Deformations are generated by a sequence of vector fields {X(2), X(3), X(4), ...}, where each X(k) is homogeneous of degree k with respect to a grading induced by rescaling. Constructing recursively the vector fields X(k), 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: usingMiura-quasitriviality, the set of linear relations expresses the tangency of the vector fields X(k) 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 Lorenzoni P (2002 J. Geom. Phys. 44 331-75), we construct the nontrivial 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.