On modified $B$KP systems and generalizations

We find the form of the Orlov–Schulman operator of the modified $B$KP hierarchy, which played a pivotal role in the construction of additional symmetries for the modified $B$KP hierarchy. We investigate the tau functions of the modified $B$KP hierarchy and give many interesting properties, including Hirota bilinear identities and $($differential$)$ Fay identities. We also present the multicomponent modified $B$KP hierarchy and define a series of additional flows of the multicomponent modified $B$KP hierarchy that constitute an $N$-fold direct product of the positive half of the quantum torus symmetries. Finally, we introduce the noncommutative modified $B$KP hierarchy and derive its symmetries, as we do for the multicomponent modified $B$KP hierarchy.