1999/07/23 by Maciej Dunajski, Paul Tod
Mathematics · Physics and Astronomy · #math.DG #gr-qc #nlin.SI #solv-int #msc:52B35 #msc:53C55
published as Differ.Geom.Appl. 14 (2001) 39-55 · 14 pages
arxiv created 1999/07/23 · arxiv updated 2009/11/30
We consider four (real or complex) dimensional hyper-Kähler metrics with a conformal symmetry K. The three-dimensional space of orbits of K is shown to have an Einstein-Weyl structure which admits a shear-free geodesics congruence for which the twist is a constant multiple of the divergence. In this case the Einstein-Weyl equations reduce down to a single second order PDE for one function. The Lax representation, Lie point symmetries, hidden symmetries and the recursion operator associated with this PDE are found, and some group invariant solutions are considered.