The Lax–Sato integrable heavenly equations on functional supermanifolds and their Lie-algebraic structure

Abstract

A Lie-algebraic approach to constructing the Lax–Sato integrable superanalogs of heavenly equations by use of the loop Lie algebra of superconformal vector fields on a 1|N-dimensional supertorus is proposed. In the framework of this approach integrable superanalogs of the Mikhalev–Pavlov heavenly equation are obtained for all $$N\in {\mathbb {N}} {\setminus } \{ 4,5 \}$$ as well as Shabat type reductions for all $$N\in {\mathbb {N}}$$. The Lax–Sato integrable superanalogs of the generalized Liouville heavenly equations are found by means of the Lie algebra of holomorphic in “spectral” parameter superconformal vector fields on a 1|N-dimensional complex supertorus.