The only nontrivial exact solutions reported to-date for the field equations of the null-surface formulation (NSF) of general relativity are for the (\(2+1\))-dimensional version of the theory, where three such solutions are known. This work presents a new family of NSF solutions. The corresponding general relativistic spacetimes are shown to span three different Petrov types, depending upon the choices that are made for various parameters. All of the scalar invariants for the spacetimes are constant, as are all of the eigenvalues of the Cotton-York tensor. The physical nature of a possible source term is discussed in detail, and two of the previously known NSF solutions are presented as special cases. The new family of solutions was derived by assuming additive separability—meaning that the dependent variable in the field equations is represented as a sum. This effectively turns the main NSF field equation (which is a partial differential equation) into an ordinary differential equation that is exactly solvable. The possibility of adapting this approach to the (\(3+1\))-dimensional version of the NSF is discussed.

迄今为止，关于广义相对论的零表面公式 (NSF) 的场方程报告的唯一非平凡精确解是 ( \(2+1\)) 维版本的理论，其中三个这样的解决方案是已知的。这项工作提出了一系列新的 NSF 解决方案。相应的广义相对论时空显示为跨越三种不同的彼得罗夫类型，这取决于对各种参数所做的选择。时空的所有标量不变量都是常数，Cotton-York 张量的所有特征值也是如此。详细讨论了可能源项的物理性质，并且将两个先前已知的 NSF 解决方案作为特殊情况介绍。新的解族是通过假设加性可分离性推导出来的——这意味着场方程中的因变量表示为和。这有效地将主要的 NSF 场方程（这是一个偏微分方程）转换为一个完全可解的常微分方程。使这种方法适应于（讨论了 NSF 的\(3+1\) ) 维版本。