Abstract
A Hamiltonian system in two dependent variables is presented with properties analogous to the Hamiltonian systems associated with the six Painlevé equations. Its solutions are meromorphic functions in the complex plane having only simple poles with three possible residues given by the third roots of unity. Like the Painlevé equations \(P_\mathrm{II}\)–\(P_\mathrm{VI}\) the system has families of rational solutions that can be obtained by applying Bäcklund transformations to certain seed solutions.
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Notes
After writing this article it was pointed out to the author by Norbert Steinmetz that the system presented here is in fact closely related to the fourth Painlevé equation. However, since then the article [16] has appeared which further studies the solutions of this system in their own right.
This lemma is quite technical; however, it forms an essential part of the proof.
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Communicated by Ilpo Laine.
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Kecker, T. A Cubic Hamiltonian System with Meromorphic Solutions. Comput. Methods Funct. Theory 16, 307–317 (2016). https://doi.org/10.1007/s40315-015-0147-6
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DOI: https://doi.org/10.1007/s40315-015-0147-6