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The Ablowitz-Ramani-Segur conjecture states that a nonlinear Partial Differential Equation is solvable by the Inverse Scattering Method only if every nonlinear Ordinary Differential Equation obtained by exact reduction has the Painlevé Property.
See also Inverse Scattering Method
References
Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley,
p. 351, 1989.