Bilinear Identities and Squared Eigenfunction Symmetries of the BCr-KP Hierarchy

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.