Virasoro symmetry of constrained KP hierarchies

Abstract The conventional formulation of additional nonisospectral sysmmetries for the full Kadomtsev-Petviashvili (KP) integrable hierarchy is not compatible with the reduction to the important class of constrained KP (cKP) integrable models. This paper solves explicitly the problem of compatibility of the Virasoro part of additional symmetries with the underlying constraints of cKP hierarchies. Our construction involves an appropriate modification of the standard additional-symmetry flows by adding a set of “ghost symmetry” flows. We also discuss the special case of cKP — truncated KP hierarchies, obtained as Darboux-Backlund orbits of initial purely differential Lax operators. Our construction establishes the condition for commutativity of the additional-symmetry flows with the discrete Darboux-Backlund transformations of cKP hierarchies leading to a new derivation of the string-equation constraint in matrix models.