We study the group properties and the similarity solutions for the constraint conditions of anti-self-dual null Khler 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.

我们研究了具有至少一个Killing对称向量的反自对偶零Khler四维流形的约束条件的群性质和相似解。具体来说，我们应用李对称理论来确定使系统保持不变的单参数点变换的所有无穷小生成元。我们使用这些变换来定义不变的相似变换，这些变换用于简化微分方程并找到时空的确切形式。我们表明约束方程允许无限数量的对称性，可用于构造无限数量的相似变换。