Self-dual Einstein spaces via a permutability theorem for the Tzitzeica equation

Abstract

The generation of integrable (differential) difference equations via a suitable reinterpretation of Bäcklund transformations and associated permutability theorems has become a standard technique in soliton theory. Here, it is shown that a permutability theorem for the classical Tzitzeica equation leads, in the natural continuum limit, to a novel fully symmetric form of the equation governing self-dual Einstein spaces in four dimensions. As a by-product of this construction, the associated linear representation is found which turns out to be the Lax pair for the self-dual Yang-Mills equations with four translational symmetries and the gauge group of volume preserving diffeomorphisms.