In this paper, we mainly investigate an equivalent form of the constrained modified KP hierarchy: the bilinear identities. By introducing two auxiliary functions ρ and σ , the corresponding identities are written into the Hirota forms. Also, we give the explicit solution forms of ρ and σ .

In this paper, we mainly investigate two kinds of gauge transformations for the constrained modified KP hierarchy in Kupershmidt-Kiso version. The corresponding gauge transformations are required to keep not only the Lax equation but also the Lax operator. For this, by selecting the special generating eigenfunction and adjoint eigenfunction, the elementary gauge transformation operators of modified KP hierarchyx , become the ones in the constrained case. Finally, the corresponding successive applications of T D and T I on the eigenfunction Φ and the adjoint eigenfunction Ψ are discussed.

BC r -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 BC r -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 BC r -KP hierarchy, and then generalize them into the constrained case. Next, we investigate the squared eigenfunction symmetries for the BC r -KP hierarchy and its constrained case, and also the connections with the additional symmetries. It is found that the constrained BC r -KP hierarchy can be defined by identifying the time flow with the squared eigenfunction symmetries.

Additional symmetry is an important kind of symmetries depending explicitly on the time and space variables, which can be expressed through Sato–Bäcklund transformations. In this paper, we construct Sato–Bäcklund transformations of the modified KP hierarchy and its constrained cases. Then the string equations of the [Formula: see text]-reduced modified KP hierarchy are established by requiring the system independent on some additional symmetry flows, which are expressed by the Lax operator [Formula: see text] and the Orlov–Shulman’s operator [Formula: see text]. At last, we obtain the negative Virasoro constraint on the two tau functions of the 2-reduced modified KP hierarchy satisfying the string equations.

In this paper, we mainly study the gauge transformations of the constrained q-deformed Kadomtsev–Petviashvili (q-KP) hierarchy. Different from the usual case, we have to consider the additional constraints on the Lax operator of the constrained q-deformed KP hierarchy, since the form of the Lax operator must be kept when constructing the gauge transformations. For this reason, the selections of generating functions in elementary gauge transformation operators [Formula: see text] and [Formula: see text] must be very special, which are from the constraints in the Lax operator. At last, we consider the successive applications of n-step of [Formula: see text] and k-step of [Formula: see text] gauge transformations.