Rational solutions for Schwarzian integrable hierarchies

We give an approach to finding rational solutions of completely intagrable hierarchies, which makes use of the relationship between modifications and the Schwarzian equations obtained via the singular manifold method. This ex tends the recent work of Kudryashov, which allowed a simple derivation of t he iteration used to construct sequences of such solutions. We also give a closed form for the index polynomial of the Schwarzian Korteweg-de Vries hi erarchy. In addition we consider the representation of rational solutions using lowe r families of the hierarchy. We give a simple representation under which th e rational solutions remain solutions of every Bow of the hierarchy. This r epresentation also allows the inclusion of arbitrary data corresponding to negative indices. We use our method to derive an alternative form of the Backlund transformat ion for the hierarchy of the second Painleve equation, as well as new solut ions of a hierarchy of breaking soliton equations. We also present here for the first time a Schwarzian version of this breaking soliton hierarchy.