Partner symmetries and non-invariant solutions of four-dimensional heavenly equations

Details Partner symmetries and non-invariant solutions of four-dimensional heavenly equations Partner symmetries and non-invariant solutions of four-dimensional heavenly equations

2004-03-12

arxiv.org/abs/math-ph/0403020v3

J.Phys. A37 (2004) 7527-7546

Physics

Mathematical Physics

20 pages, 1 table, corrected typos

Scientific paper

10.1088/0305-4470/37/30/010

We extend our method of partner symmetries to the hyperbolic complex Monge-Amp\`ere equation and the second heavenly equation of Pleba\~nski. We show the existence of partner symmetries and derive the relations between them for both equations. For certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to systems of linear equations by an appropriate Legendre transformation. The solutions of these linear equations are generically non-invariant. As a consequence we obtain explicitly new classes of heavenly metrics without Killing vectors.

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