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We extend the Mason-Newman Lax pair for the elliptic complex Monge-Amp\`ereequation so that this equation itself emerges as an algebraic consequence. Weregard the function in the extended Lax equations as a complex potential. Weidentify the real and imaginary parts of the potential, which we call partnersymmetries, with the translational and dilatational symmetry characteristicsrespectively. Then we choose the dilatational symmetry characteristic as thenew unknown replacing the K\"ahler potential which directly leads to a Legendretransformation and to a set of linear equations satisfied by a single realpotential. This enables us to construct non-invariant solutions of the Legendretransform of the complex Monge-Amp\`ere equation and obtain hyper-K\"ahlermetrics with anti-self-dual Riemann curvature 2-form that admit no Killingvectors.Comment: submitted to J. Phys.

Partner symmetries of the complex Monge–Amp re equation yield hyper-K hler metrics without continuous symmetries