### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 043, 13 pages      arXiv:1005.0153      https://doi.org/10.3842/SIGMA.2011.043
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

### Recursions of Symmetry Orbits and Reduction without Reduction

Andrei A. Malykh a and Mikhail B. Sheftel b
a) Department of Numerical Modelling, Russian State Hydrometeorlogical University, Malookhtinsky pr. 98, 195196 St. Petersburg, Russia
b) Department of Physics, Bogazici University 34342 Bebek, Istanbul, Turkey

Received January 29, 2011, in final form April 25, 2011; Published online April 29, 2011

Abstract
We consider a four-dimensional PDE possessing partner symmetries mainly on the example of complex Monge-Ampère equation (CMA). We use simultaneously two pairs of symmetries related by a recursion relation, which are mutually complex conjugate for CMA. For both pairs of partner symmetries, using Lie equations, we introduce explicitly group parameters as additional variables, replacing symmetry characteristics and their complex conjugates by derivatives of the unknown with respect to group parameters. We study the resulting system of six equations in the eight-dimensional space, that includes CMA, four equations of the recursion between partner symmetries and one integrability condition of this system. We use point symmetries of this extended system for performing its symmetry reduction with respect to group parameters that facilitates solving the extended system. This procedure does not imply a reduction in the number of physical variables and hence we end up with orbits of non-invariant solutions of CMA, generated by one partner symmetry, not used in the reduction. These solutions are determined by six linear equations with constant coefficients in the five-dimensional space which are obtained by a three-dimensional Legendre transformation of the reduced extended system. We present algebraic and exponential examples of such solutions that govern Legendre-transformed Ricci-flat Kähler metrics with no Killing vectors. A similar procedure is briefly outlined for Husain equation.

Key words: Monge-Ampère equation; partner symmetries; symmetry reduction; non-invariant solutions; anti-self-dual gravity; Ricci-flat metric.

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References

1. Sheftel M.B., Malykh A.A., On classification of second-order PDEs possessing partner symmetries, J. Phys. A: Math. Theor. 42 (2009), 395202, 20 pages, arXiv:0904.2909.
2. Malykh A.A., Nutku Y., Sheftel M.B., Partner symmetries of the complex Monge-Ampère equation yield hyper-Kähler metrics without continuous symmetries, J. Phys. A: Math. Gen. 36 (2003), 10023-10037, math-ph/0305037.
3. Malykh A.A., Nutku Y., Sheftel M.B., Partner symmetries and non-invariant solutions of four-dimensional heavenly equations, J. Phys. A: Math. Gen. 37 (2004), 7527-7545, math-ph/0403020.
4. Malykh A.A., Nutku Y., Sheftel M.B., Lift of noninvariant solutions of heavenly equations from three to four dimensions and new ultra-hyperbolic metrics, J. Phys. A: Math. Theor. 40 (2007), 9371-9386, arXiv:0704.3335.
5. Sheftel M.B., Malykh A.A., Lift of invariant to non-invariant solutions of complex Monge-Ampère equations, J. Nonlinear Math. Phys. 15 (2008), suppl. 3, 375-385, arXiv:0802.1463.
6. Plebañski J.F., Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975), 2395-2402.
7. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
8. Doubrov B., Ferapontov E.V., On the integrability of symplectic Momge-Ampère equations, J. Geom. Phys. 60 (2010), 1604-1616, arXiv:0910.3407.
9. Dunajski M., Mason L.J., Twistor theory of hyper-Kähler metrics with hidden symmetries, J. Math. Phys. 44 (2003), 3430-3454, math.DG/0301171.
10. Griffiths P., Harris J., Principles of algebraic geometry, Wiley & Sons, Inc., New York, 1994.
11. Malykh A.A., Sheftel M.B., General heavenly equation governs anti-self-dual gravity, J. Phys. A: Math. Theor. 44 (2011), 155201, 11 pages, arXiv:1011.2479.
12. Atiyah M.F., Hitchin N.J., Singer I.M., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425-461.
13. Dunajski M., Solitons, instantons, and twistors, Oxford Graduate Texts in Mathematics, Vol. 19, Oxford University Press, Oxford, 2010.
14. Husain V., Self-dual gravity as a two dimensional theory and conservation laws, Classical Quantum Gravity 11 (1994), 927-937, gr-qc/9310003.