ar X iv : m at h - ph / 0 40 30 20 v 3 2 8 Ju l 2 00 4 Partner symmetries and non - invariant solutions of four - dimensional heavenly equations

0 40 30 20 v 1 1 2 M ar 2 00 4 Partner symmetries and non - invariant solutions of four - dimensional heavenly equations

We extend our method of partner symmetries to the hyperbolic complex Monge-Ampère equation and the second heavenly equation of Plebañski. 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 syst...

v 2 3 0 M ar 2 00 4 Partner symmetries and non - invariant solutions of four - dimensional heavenly equations

We extend our method of partner symmetries to the hyperbolic complex Monge-Ampère equation and the second heavenly equation of Plebañski. 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 syst...

ar X iv : g r - qc / 0 10 50 88 v 1 2 4 M ay 2 00 1 A family of heavenly metrics

The antecedent of Einstein field equations with Euclidean signature is the complex elliptic Monge-Ampère equation, CMA2. For elliptic CMA2 solutions determine hyper-Kähler, self-dual and therefore Ricci-flat metrics with Euclidean signature. We shall consider a symmetry reduction of CMA2 to solutions that admit only a single Killing vector. Systematic studies of vacuum metrics with one Killing ...

X iv : m at h - ph / 0 30 50 37 v 2 2 5 Ju l 2 00 3 Partner symmetries of the complex Monge - Ampère equation yield hyper - Kähler metrics without continuous symmetries

We extend the Mason-Newman Lax pair for the elliptic complex MongeAmpère equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. They imply the determining equation for symmetries of the complex Monge-Ampère equation as their differential compatibility condition. We shall identify the real and imagin...

ar X iv : m at h - ph / 0 60 70 40 v 1 2 0 Ju l 2 00 6 Invariant Form of BK - factorization and its Applications