Bäcklund transformation and the Painlevé property

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Abstract
When a differential equation possesses the Painlevé property it is possible (for specific equations) to define a Bäcklund transformation (by truncating an expansion about the ‘‘singular’’ manifold at the constant level term). From the Bäcklund transformation, it is then possible to derive the Lax pair, modified equations and Miura transformations associated with the ‘‘completely integrable’’ system under consideration. In this paper completely integrable systems are considered for which Bäcklund transformations (as defined above) may not be directly defined. These systems are of two classes. The first class consists of equations of Toda lattice type (e.g., sine–Gordon, Bullough–Dodd equations). We find that these equations can be realized as the ‘‘minus‐one’’ equation of sequences of integrable systems. Although the ‘‘Bäcklund transformation’’ may or may not exist for the ‘‘minus‐one’’ equation, it is shown, for specific sequences, that the Bäcklund transformation does exist for the ‘‘positive’’ equations of the sequence. This, in turn, allows the derivation of Lax pairs and the recursion operation for the entire sequence. The second class of equations consists of sequences of ‘‘Harry Dym’’ type. These equations have branch point singularities, and, thus, do not directly possess the Painlevé property. Yet, by a process similar to the ‘‘uniformization’’ of algebraic curves, their solutions may be parametrically’’ represented by ‘‘meromorphic’’ functions. For specific systems, this is shown to provide a natural extension of the Painlevé property.