Abstract
A decade ago the impressive success of the Painleve test in locating integrable systems within parameterized families of ordinary differential equations led to speculation, by some, that passing the test was a necessary condition for integrability. A subsequent infinite list of examples produced by A. Ramani et. al. [R-D-G] forced a qualification of this conjecture, and counterexamples to that qualification have now appeared (e.g. see [K-CR]). With the benefit of hindsight it seems fair to say that the test is more appropriately associated with ‘algebraic’ integrability (see [A-vM2, vMo]) and, more generally, with the possibility of augmenting isoenergy surfaces or the phase space so as to realize the flow within a complete C-action. In the algebraically integrable case this augmentation is most elegantly accomplished by means of the spectral curve of a Lax pair. (We do not assume Lax pairs and associated spectral curves are familiar to the reader.) Unfortunately, Lax pairs admitting useful spectral curves can be difficult to produce.
Department of Mathematics, Hunter College, 695 Park Ave., New York, NY 10021. Author’s research supported in part by NSF Grant # DMS8802911.
Department of Mathematics, University of Calgary, Calgary, Alberta T2N 1N4 Canada. Author’s research supported in part by the Natural Sciences and Engineering Research Council of Canada.
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Churchill, R.C., Falk, G.T. (1995). Lax Pairs in the Henon-Heiles and Related Families. In: Dumas, H.S., Meyer, K.S., Schmidt, D.S. (eds) Hamiltonian Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol 63. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8448-9_5
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