Based on the Lax operator L and Orlov-Shulman's M operator,the string equations of the q-KP hierarchy are established from the special additional symmetry flows,and the negative Virasoro constraint generators {L-n,n ≥ 1} of the 2-reduced q-KP hierarchy are also obtained.
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.
