The Symmetries And The Recursion Operators For Several Classical Integrable Hierarchies

The symmetries and the recursion operators for several kinds of classi-cal integrable hierarchies are studied: 1) the spectral representation of theKadomtsev-Petviashvili hierarchy of B type (BKP hierarchy) and the squaredeigenfunction potential of B type (BSEP) are given, then based upon theseresults, the"ghost"symmetry for the BKP hierarchy is systematically con-structed, and some applications in the derivations of the bilinear identity forthe constrained BKP hierarchy and the Adler-Shiota-van-Moerbeke (ASvM)formula for the BKP hierarchy are considered; 2) the additional symmetriesfor the Toda lattice hierarchies of B type and C type (BTL and CTL hierar-chies) are constructed, then its algebraic structures are showed as w∞B×w∞Band w∞C×w∞C respectively, and also the generating functions of the corre-ponding additional symmetries are discussed; 3) a unified construction ofthe recursion operators from the Lax representation for three integrable hi-erarchies: KP, modified KP (mKP) and Harry-Dym (HD) hierarchies undern-reduction is given; 4) a general formula of the ?ow equtions for the HDhierarchy is derived.