The N = 2 supersymmetric heavenly equation and its super-hydrodynamical reduction

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 analyse 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.