The BCr-KP hierarchy is an important sub hierarchy of the KP hierarchy, which includes the BKP and CKP hierarchies as the special cases. Some properties of the BCr-KP hierarchy and its constrained case are investigated in this paper, including bilinear identities and squared eigenfunction symmetries. We firstly discuss the bilinear identities of the BCr-KP hierarchy, and then generalize them into the constrained case. Next, we investigate the squared eigenfunction symmetries for the BCr-KP hierarchy and its constrained case, and also the connections with the additional symmetries. It is found that the constrained BCr-KP hierarchy can be defined by identifying the time flow with the squared eigenfunction symmetries.

中文翻译：

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BCr-KP 层次结构的双线性恒等式和平方本征函数对称性

BCr-KP 层次结构是 KP 层次结构的一个重要子层次结构，其中包括 BKP 和 CKP 层次结构作为特例。本文研究了 BCr-KP 层次结构及其约束情况的一些性质，包括双线性恒等式和平方本征函数对称性。我们首先讨论 BCr-KP 层次结构的双线性恒等式，然后将它们推广到受约束的情况。接下来，我们研究 BCr-KP 层次结构及其约束情况的平方特征函数对称性，以及与附加对称性的联系。发现约束 BCr-KP 层次结构可以通过识别具有平方特征函数对称性的时间流来定义。