Einstein-Weyl structures from hyper-Kähler metrics with conformal Killing vectors (Q5931425)

Four-dimensional hyper-Kähler spaces with a conformal symmetry \(K\) can be studied in terms of Einstein-Weyl structures on the space of trajectories of \(K\). For example, when \(K\) is a Killing vector, \textit{C. P. Boyer} and \textit{J. D. Finley} [J. Math. Phys. 23, 1126-1130 (1982; Zbl 0484.53051)] and \textit{R. S. Ward} [Classical Quantum Gravity 7, L95--L98 (1990; Zbl 0687.53044)] showed that the Einstein-Weyl quotient spaces can be described by the SU(\(\infty\)) Toda equation. Another example is when \(K\) is a triholomorphic homothety, that has been treated by \textit{G. Chave, P. Tod} and \textit{G. Valent} [Phys. Lett. B 383, 262-270 (1996)]. NEWLINENEWLINENEWLINEIn this paper, the authors consider the most general case of \(K\) being a conformal symmetry, non-triholomorphic Killing vector. After various definitions and formulae, the canonical form of an allowed conformal Killing vector is given. Then the authors obtain explicit nonlinear differential equations which govern the quotient Einstein-Weyl spaces, and give the Lax representation of the reduced equations. NEWLINENEWLINENEWLINEIn sections 6-8 the authors find and classify the Lie point symmetries and obtain explicit solutions, study hidden symmetries and a recursion operator for its generation, and show that the Einstein-Weil quotient spaces are characterized as those that admit a shear free geodesic congruence for which twist and divergence are linearly dependent [see also the work by \textit{D. M. J. Calderbank} and \textit{H. Pedersen}, Ann. Inst. Fourier 50, No. 3, 921-963 (2000; Zbl 0970.53027)].