Einstein–Weyl structures from Hyper–Kähler metrics with conformal Killing vectors

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. 1 Three-dimensional Einstein–Weyl spaces Three-dimensional Einstein–Weyl (EW) geometries were first considered by Cartan [3] and then rediscovered by Hitchin [8] in the context of twistor theory. They constitute an interesting generalisation of (the otherwise locally trivial) Einstein condition in three dimensions. In this paper we shall consider four-dimensional anti-self-dual (ASD) vacuum (or complexified hyper-Kähler) spaces with a conformal symmetry. By a general construction [9] such spaces will give rise to Einstein–Weyl structures on the space of trajectories of the given conformal symmetry K. The cases where K is a pure Killing vector or a tri-holomorphic homothety have been extensively studied [1, 15, 4, 10]. Therefore we shall consider the most general case ofK being a conformal, nontriholomorphic Killing vector. We begin by collecting various definitions and formulae concerning three-dimensional Einstein–Weyl spaces (see [11] for a fuller account). In the next section we shall give the canonical form of an allowed conformal Killing vector in a natural coordinate system associated with the Kähler potential. Then we shall look at solutions to a non-linear Monge– Ampere equation (the so called ‘first heavenly equation’ [12]) (2.7) for the Kähler potential which admit the symmetry K. This will give rise to a new integrable system in three dimensions and to the corresponding EW geometries. In Section 3 we shall give the Lax representation of the reduced equations. When Euclidean reality conditions are imposed (Section 5) we shall recover some known results [1, 15] as limiting cases of our construction. In Section 6 we shall find and classify the Lie point symmetries of the field equations in three dimensions (and so the Killing vectors of the associated Weyl structure), and consider some group invariant solutions. In Section 7 we shall study hidden symmetries and the recursion operator associated to the three-dimensional system. In Section 8 we shall show that the EW structures studied in this paper admit a shear free geodesic congruence for which twist and divergence are linearly dependent. Let W be an n-dimensional complex manifold, with a torsion-free connection D and a conformal metric [h]. We shall call W a Weyl space if the null geodesics of [h] are also geodesics for D. This condition is equivalent to Dihjk = νihjk (1.1) ∗email: [email protected]