Exact Poisson pencils, -structures and topological hierarchies

We discuss, in the framework of the Dubrovin–Zhang perturbative approach to integrable evolutionary PDEs in 1 + 1 dimensions, the role of a special class of Poisson pencils, called exact Poisson pencils. In particular we show that, in the semisimple case, exactness of the pencil is equivalent to the constancy of the so-called “central invariants” of the theory that were introduced by Dubrovin, Liu and Zhang.