\(\xi \)-submanifolds and \(\xi \)-translators are, respectively, the natural generalizations of self-shrinkers and translators of the mean curvature flow and, in the case of codimension one, they are previously known as \(\lambda \)-hypersurfaces and \(\lambda \)-translators, respectively. In this paper, we study the complete Lagrangian space-like \(\xi \)-surfaces and \(\xi \)-translators in \({\mathbb R}^4_2\), the pseudo-Euclidean 4-spaces of signature 2 endowed with the canonical complex structure. As the result, we first obtain a classification theorem for all complete Lagrangian space-like \(\xi \)-surfaces in \({\mathbb R}^4_2\) of constant square norm of the second fundamental form. Then the main idea of the proof also allows us to obtain a similar classification theorem for \(\xi \)-translators in \({\mathbb R}^4_2\) by a Bernstein-type theorem for space-like translators in a general pseudo-Euclidean space \({\mathbb R}^{m+p}_p\), which is of independent significance.

\（\ xi \）-子流形和\（\ xi \） -平移器分别是自收缩器和平均曲率流平移器的自然概括，在余维数为1的情况下，它们以前被称为\ （\ lambda \）- hypersurfaces和\ （\ lambda \）- translators 。在本文中，我们研究完整的拉格朗日空间状\（\ XI \） -surfaces和\（\ XI \） -translators在\（{\ mathbb R} ^ 4_2 \） ，伪欧几里得4-位签名2具有规范的复杂结构。结果，我们首先获得所有完备的拉格朗日空间类\（\ xi \）-曲面的分类定理。第二个基本形式的恒定平方范数的\（{{mathbb R} ^ 4_2 \）。然后，证明的主要思想也可以让我们得到一个类似的分类定理\（\ XI \） -translators在\（{\ mathbb R} ^ 4_2 \）由伯恩斯坦型定理为空间像在翻译具有独立意义的一般伪欧几里德空间\（{\ mathbb R} ^ {m + p} _p \）。