# Symmetries, integrals and hierarchies of new (3+1)-dimensional bi-Hamiltonian systems of Monge--Amp\`ere type

## TL;DR

This paper explores new (3+1)-dimensional bi-Hamiltonian systems of Monge-Ampère type, analyzing their symmetries, conserved quantities, and hierarchies, with implications for gravitational instantons without Killing vectors.

## Contribution

It introduces four new bi-Hamiltonian heavenly systems in 3+1 dimensions and highlights the role of inverse recursion operators in their hierarchy and symmetry analysis.

## Key findings

- Identification of four new bi-Hamiltonian systems
- Role of inverse recursion operators in hierarchy construction
- Generation of nonlocal symmetry flows leading to special gravitational metrics

## Abstract

We study point symmetries, the corresponding conserved densities and hierarchies of four new bi-Hamiltonian heavenly systems in 3+1 dimensions which we discovered recently. We exhibit an important role played by the inverse recursion operators in the description of the hierarchies. Their use is twofold, either to provide the correct bi-Hamiltonian representation or to generate nonlocal symmetry flows. Invariant solutions w.r.t. nonlocal symmetries will generate (anti-)self-dual gravitational metrics which do not admit Killing vectors which is a characteristic feature of $K3$ gravitational instanton.