  There are investigated the Lie algebraic structure and integrability
properties of a very interesting class of nonlinear dynamical systems called
the heavenly equations, which were initiated by Pleba\'nski and later analyzed
in a series of articles. Based on the AKS-algebraic and related R-structure
schemes there are studied orbits of the corresponding co-adjoint actions,
deeply related with the classical Lie-Poisson type structures on them. There is
successively demonstrated that their compatibility condition proves to coincide
exactly with the corresponding heavenly equation under regard. It is stated
that all the heavenly equations allow such an origin and can be equivalently
represented as a Lax type compatibility condition for specially built loop
vector fields on the torus. The infinite hierarchy of conservations laws,
related with the heavenly equations is discussed, its analytical structure,
connected with the Casimir invariants is mentioned. In addition, the typical
examples of the heavenly equations, demonstrating in details their
integrability within the scheme devised in the work, are presented. The
Lagrangian representation of the heavenly equations following from their
Hamiltonicity with respect to both related commuting to each other evolution
flows, as well as their associated bi-Hamiltonian structure are also discussed.
The relationship of the heavenly type dynamical ssytems with the classical
Lagrange-d'Alambert principle is demonstrated.
