New Lax pair for 2D dimensionally reduced gravity (Q2716901)

When one searches for closed-form solutions of the 4-dimensional Einstein equation, then it proves useful to assume some symmetries. Under the assumption that the isometries have 2-dimensional orbits, one can reformulate the problem to a problem within 2 dimensions. One of the many ways to do that is deduced in [\textit{H.-J. Schmidt}, Int. J. Mod. Phys. D 7, 215-223 (1998; Zbl 0946.83005)].NEWLINENEWLINENEWLINEIn the present paper the authors re-analyze this problem for the dimensionally reduced Einstein equation such that the Ernst equation appears. They concentrate on a new version of the Lax pair based on the affine \(\text{SL}(2,\mathbb{R})\) Kac-Moody algebra, and they give the corresponding Poisson brackets in many details.