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

Abstract

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 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.

20 pages, 1 table, corrected typos

References in corpus (6)

• Hyper-K{ä}hler Hierarchies and their twistor theory
• Twistor theory of hyper-K{ä}hler metrics with hidden symmetries
• Supersymmetric sigma-models, twistors, and the Atiyah-Hitchin metric
• Gravitational instantons admit hyper-Kähler structure
• Partner symmetries of the complex Monge-Ampere equation yield hyper-Kahler metrics without continuous symmetries
• Anti-self-dual Riemannian metrics without Killing vectors, can they be realized on K3?

Cited by in corpus (11)

• A Simple Construction of Recursion Operators for Multidimensional Dispersionless Integrable Systems
• Multi-Hamiltonian structure of Plebanski's second heavenly equation
• Integrable dispersionless PDE in 4D, their symmetry pseudogroups and deformations
• On a method for constructing the Lax pairs for nonlinear integrable equations
• Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation
• On a class of second-order PDEs admitting partner symmetries
• Bi-Hamiltonian representation, symmetries and integrals of mixed heavenly and Husain systems
• Lift of noninvariant solutions of heavenly equations from three to four dimensions and new ultra-hyperbolic metrics
• Self-dual gravity is completely integrable
• SDiff(2) and uniqueness of the Plebański equation
• Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations