Exact Anti-Self-Dual four-manifolds with a Killing symmetry by similarity transformations

We study the group properties and the similarity solutions for the constraint conditions of anti-self-dual null K\"{a}hler four-dimensional manifolds with at least a Killing symmetry vector. Specifically we apply the theory of Lie symmetries to determine all the infinitesimal generators of the one-parameter point transformations which leave the system invariant. We use these transformations to define invariant similarity transformations which are used to simplify the differential equations and find the exact form of the spacetime. We show that the constraint equations admit an infinite number of symmetries which can be used to construct an infinite number of similarity transformations.