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 extends the recent work of Kudryashov, which allowed a simple derivation of the iteration used to construct sequences of such solutions. We also give a closed form for the index polynomial of the Schwarzian Korteweg-de Vries hierarchy. In addition we consider the representation of rational solutions using lower families of the hierarchy. We give a simple representation under which the rational solutions remain solutions of every flow of the hierarchy. This representation also allows the inclusion of arbitrary data corresponding to negative indices. We use our method to derive an alternative form of the Bäcklund transformation for the hierarchy of the second Painlevé equation, as well as new solutions of a hierarchy of breaking soliton equations. We also present here for the first time a Schwarzian version of this breaking soliton hierarchy.