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We propose a generalized super Camassa-Holm equation, which is completely integrable in the sense of admitting of a Lax pair and a bi-Hamiltonian structure. Through Dirac reduction, we obtain a bi-Hamiltonian structure of the super Camassa-Holm equation introduced by Geng, Xue and Wu [Stud. Appl. Math. 130 (2013) 1-16]. By introducing an appropriate reciprocal transformation, we connect the generalized super Camassa-Holm equation with a negative generalized super KdV equation, which is also equipped with a Lax pair and a bi-Hamiltonian structure. [ABSTRACT FROM AUTHOR]

Abstract: The changing rule of super Hamiltonian operators under general superconformal transformations is established by means of investigating behavior of supersymmetric Euler derivatives under the same kind of changes of variables. In the two particular yet frequent cases such as supersymmetric Miura-type transformations and reciprocal transformations, the results are detailed and applied to construct bi-Hamiltonian structures of some supersymmetric evolution equations. As an interesting example, one of supersymmetric Harry Dym equations is shown to be a bi-Hamiltonian system through its reciprocal link to the classical Harry Dym equation. [Copyright &y& Elsevier]