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Supersymmetric Kuper Camassa-Holm Equation and Geodesic Flow : A Novel Approach
We use the logarithmic $2$-cocycle and the action of $Vect(S^1)$ on the space of Pseudo-differential symbols to derive one particular type of supersymmetric KdV equation, known as Kuper-KdV equation. This equation was formulated by Kupershmidt and it is different from the Manin-Radul-Mathieu type equation. The two Super KdV equations behave differently under a supersymmetric transformation and Kupershmidt version does not preserve SUSY transformation. In this paper we study the second type of supersymmetric generalization of the Camassa-Holm equation correspoding to Kuper-KdV equation via standard embedding of super vector fields into the Lie algebra of graded peudodifferential symbols. The natural lift of the action of superconformal group $SDiff$ yields $SDiff$ module. This method is particularly useful to construct Moyal quantized systems.