Poisson structures of some heavenly type dynamical systems

Abstract

The paper investigates the Poisson structures associated with dynamical systems of the heavenly type, focusing on the Mikhalev-Pavlov and Pleba\'nski equation. The dynamical system is represented as a Hamiltonian system on a functional manifold, and Poisson brackets are defined based on a non-degenerate Poisson operator. The study explores the Lax-type integrability and bi-Hamiltonian properties of the systems, revealing the existence of compatible Poisson operators. The Lie-algebraic approach, particularly the AKS-algebraic and $\mathcal{R}$-structure schemes, is employed to analyze the holomorphic loop Lie algebra, providing insights into the Lie-algebraic structure of heavenly equations. The Mikhalev-Pavlov and Pleba\'nski equations are studied in detail, and the associated Poisson brackets for specific coordinate functions are derived, revealing interesting mathematical properties. The paper establishes a foundation for understanding the symplectic structures associated with heavenly-type dynamical systems.