ABSTRACT
This review is devoted to the Buhl compatible vector field equation problem, emphasizing its Pfeiffer and Lax–Sato type solutions. We analyze the related Lie-algebraic structures and integrability of the heavenly equations. AKS-algebraic and related R $ \mathcal R $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429470462/301849f2-53e7-4646-acba-6259bf733f01/content/inline-math2294.tif"/> -structure schemes are used to study the corresponding co-adjoint actions. Their compatibility conditions are shown to coincide with the corresponding heavenly equations, all of which originate in this way and can be represented as a Lax compatibility condition. The infinite hierarchy of conservation laws for the heavenly equations is described along with its Casimir invariant connection and several examples are presented. An interesting related Lagrange–d’Alembert principle is also discussed. A generalization of the scheme, related to the loop Lie superalgebra of the Lie super group of superconformal diffeomorphisms of the 1|N-dimensional supertorus, is used to construct Lax–Sato integrable supersymmetric analogs of the Mikhalev–Pavlov heavenly equation for every N ∈ N \ { 4 ; 5 } $ N\in \mathbb N \backslash \{4;5\} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429470462/301849f2-53e7-4646-acba-6259bf733f01/content/inline-math2295.tif"/> . Super-analogs of Liouville equations are constructed using superconformal maps.
