The classical M. A. Buhl problem, its Pfeiffer – Sato solutions and the classical Lagrange – D’Alembert principle for the integrable heavenly type nonlinear equations - Prykarpatsky, Ya. A.; Samoilenko, A. M. (Ukrains’kyi Matematychnyi Zhurnal, 2017)
The survey is devoted to old and recent investigations of the classical M. A. Buhl problem of description of the compatiblelinear vector field equations and their general M. G. Pfeiffer and modern Lax – Sato-type special solutions. In particular,we analyze the related Lie-algebraic structures and the properties of integrability for a very interesting class of nonlineardynamical systems called the dispersion-free heavenly type equations, which were introduced by Pleba´nski and lateranalyzed in a series of articles. The AKS-algebraic and related \scrR -structure schemes are used to study the orbits of thecorresponding coadjoint actions, which are intimately connected with the classical Lie – Poisson structures on them. It isshown that their compatibility condition coincides with the corresponding heavenly type equations under consideration.It is also demonstrated that all these equations are originated in this way and can be represented as a Lax compatibilitycondition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws relatedto the heavenly equations is described and its analytic structure connected with the Casimir invariants is indicated. Inaddition, we present typical examples of equations of this kind demonstrating in detail their integrability via the schemeproposed in the paper. The relationship between a very interesting Lagrange – d’Alembert-type mechanical interpretationof the devised integrability scheme and the Lax – Sato equations is also discussed.