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 imaginary parts of the potential, which we call partner symmetries, with the translational and dilatational symmetry characteristics respectively. Then we choose the dilatational symmetry characteristic as the new unknown replacing the Kähler potential. This directly leads to a Legendre transformation. Studying the integrability conditions of the Legendre-transformed system we arrive at a set of linear equations satisfied by a single real potential. This enables us to construct non-invariant solutions of the Legendre transform of the complex Monge-Ampère equation. Using these solutions we obtained explicit Legendre-transformed hyperKähler metrics with anti-self-dual Riemann curvature 2-form that admit no Killing vectors. They satisfy the Einstein field equations with Euclidean signature. We give the detailed derivation of the solution announced earlier and present a new solution with an added parameter. We compare our method of partner symmetries for finding non-invariant solutions to that of Dunajski and Mason who use ‘hidden’ symmetries for the same purpose. PACS numbers: 04.20.Jb, 02.40.Ky 2000 Mathematical Subject Classification: 35Q75, 83C15