From a bi‐Hamiltonian viewpoint the equivalence of two supersymmetric Korteweg–deVries theories, introduced by Manin–Radul and Laberge–Mathieu, is discussed herein. It is shown that the transformation connecting the two theories (proposed recently in the literature) preserves the bi‐Hamiltonian structures; moreover, another derivation of this transformation, stemming from bi‐Hamiltonian reduction theory and strongly emphasizing the geometrical meaning of the above equivalence is presented.

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Strictly speaking, the supermatrices appearing in the linear systems (4.8) and (4.10) are not σ(m) and σ(m), but more precisely two representatives in the corresponding equivalence classes. The approach of Ref. 5 removes this technical problem working directly with gl(2,2) instead of A(1,1). The Lie superalgebra gl(2,2) consists of all (2/2)×(2/2) supermatrices, without any equivalence relation.
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