2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/184582We extend the Mason-Newman Lax pair for the elliptic complex Monge-Ampère equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. We 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 which directly leads to a Legendre transformation and to 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 and obtain hyper-Kähler metrics with anti-self-dual Riemann curvature 2-form that admit no Killing vectors.submitted to J. Phys. AMathematical PhysicsGeneral Relativity and Quantum CosmologyHigh Energy Physics - TheoryDifferential GeometryExactly Solvable and Integrable Systems35Q75, 83C15text