The N=2 Supersymmetric Heavenly Equation and Its Super-Hydrodynamical Reduction

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Manifest N=2 supersymmetric Toda systems are constructed from the sl(n,n+1) superalgebras by taking into account their complex structure. In the n→ ∞ continuum limit an N=2 extension of the (2+1)-dimensional heavenly equation is obtained. The integrability is guaranteed by the existence of a supersymmetric Lax pair. We further analyze the properties of the (1+1)-dimensionally reduced system. Its bosonic sector is of hydrodynamical type. This is not the case for the whole supersymmetric system which, however, is super-hydrodynamical when properly expressed in terms of a supergeometry involving superfields and fermionic derivatives.