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

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