  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.
