Deformations of non-semisimple Poisson pencils of hydrodynamic type

We study the deformations of two-component non-semisimple Poisson pencils of hydrodynamic type associated with Balinskǐ-Novikov algebras. We show that in most cases the second order deformations are parametrized by two functions of a single variable. We find that one function is invariant with respect to the subgroup of Miura transformations, preserving the dispersionless limit, and another function is related to a one-parameter family of truncated structures. In two exceptional cases the second order deformations are parametrized by four functions. Among these two are invariants and two are related to a two-parameter family of truncated structures. We also study the lift of the deformations of n-component semisimple structures. This example suggests that deformations of non-semisimple pencils corresponding to the lifted invariant parameters are unobstructed.