The Painlevé analysis of a generic multiparameter N=2 extension of the Korteweg–de Vries (KdV) equation is presented. Unusual aspects of the analysis, pertaining to the presence of two fermionic fields, are emphasized. For the general class of models considered, we find that the only ones which manifestly pass the test are precisely the four known integrable supersymmetric KdV equations, including the SKdV1 case.

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Notice that in Ref. 3, the following system has also been shown to have the Painlevé property ut=−uxxx+6uux−3ξξxx,ξt=−ξxxx+6(uξ)x. However, the change of variable v=u−ξ∂x−1ξ transforms it into [cf.
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Notice that even if the Painlevé analysis breaks the O(2) invariance, we are still free to choose the sign of k0. Clearly, in a formulation in terms of the redefined fields ξ(±)(1)±iξ(2), we would be free to take either ξ0(+) or ξ0(−) as an arbitrary function, the other being null.
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