# Integrability of anti-self-dual vacuum Einstein equations with nonzero cosmological constant: an infinite hierarchy of nonlocal conservation laws

## TL;DR

This paper develops an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable PDE related to anti-self-dual vacuum Einstein equations with a nonzero cosmological constant, using a nonisospectral Lax pair.

## Contribution

It introduces a novel hierarchy of conservation laws and constructs an infinite-dimensional differential covering for the Przanowski equation, advancing understanding of its integrability properties.

## Key findings

- Established an infinite hierarchy of nonlocal conservation laws.
- Constructed a nonisospectral Lax pair for the Przanowski equation.
- Derived an infinite-dimensional differential covering.

## Abstract

We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable second-order PDE locally equivalent to anti-self-dual vacuum Einstein equations with nonzero cosmological constant. The hierarchy in question is constructed using a nonisospectral Lax pair for the equation under study. As a byproduct, we obtain an infinite-dimensional differential covering over the Przanowski equation.