On the c-k constrained KP and BKP hierarchies: the Fermionic pictures, solutions and additional symmetries

In this paper, we study two generalized constrained integrable hierarchies, which are called the c-k constrained KP and BKP hierarchies. The Fermionic picture of the c-k constrained KP hierarchy is given. We give some solutions for the c-k constrained KP hierarchy by using the free Fermion operators and define its additional symmetries. Its additional flows form a subalgebra of the Virasoro algebra. Furthermore, the additional flows acting on eigenfunctions qᵢ(t) and adjoint eigenfunctions rᵢ(t) of the c-k constrained KP hierarchy are presented. Next, we define the c-k constrained BKP hierarchy and obtain its bilinear identity and solutions. The algebra formed by the additional symmetric flow of the c-k constrained BKP hierarchy that we defined is still a subalgebra of the Virasoro algebra and it is a subalgebra of the algebra formed by the additional flows of the c-k constrained KP hierarchy.