Abstract. Based on the Orlov and Shulman's M operator, the additional symmetries and the string equation of the CKP hierarchy are established, and then the higher order constraints on L l are obtained. In addition, the generating function and some properties are also given. In particular, the additional symmetry flows form a new infinite dimensional algebra W C 1+∞ , which is a subalgebra of W 1+∞ .

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Abstract. In this paper, with the help of the S function and ghost symmetry for the discrete KP hierarchy which is a semi-discrete version of the KP hierarchy, the ghost flow on its eigenfunction(adjoint eigenfunction) and the spectral representation of its Baker-Akhiezer function and adjoint Baker-Akhiezer function are derived. From these observations above, some important distinctions between the discrete KP hierarchy and KP hierarchy are shown. Also we give the ghost flow on the tau function and another kind of proof of the ASvM formula of the discrete KP hierarchy.

In this paper, we construct the additional symmetries for the Toda lattice (TL) hierarchies of B type and C type (the BTL and CTL hierarchies), and show their algebraic structures are w∞ respectively. And also we discuss the generating functions of the additional symmetries.

Abstract. The additional symmetries of the constrained CKP (cCKP) and BKP (cBKP) hierarchies are given by their actions on the Lax operators, and their actions on the eigenfunction and adjoint eigenfunction {Φ i , Ψ i } are presented explicitly. Furthermore, we show that acting on the space of the wave operator, ∂ * k forms new centerless W cC 1+∞ and W cB 1+∞ -subalgebra of centerless W 1+∞ respectively. In order to define above symmetry flows ∂ * k of the cCKP and cBKP hierarchies, two vital operators Y k are introduced to revise the additional symmetry flows of the CKP and BKP hierarchies.