The conventional formulation of additional nonisospectral symmetries f
or the full Kadomtsev-Petviashvili (KP) integrable hierarchy is not co
mpatible with the reduction to the important class of constrained KP (
cKP) integrable models. This paper solves explicitly the problem of co
mpatibility of the Virasoro part of additional symmetries with the und
erlying constraints of cKP hierarchies. Our construction involves an a
ppropriate modification of the standard additional-symmetry Rows by ad
ding a set of ''ghost symmetry'' Rows. We also discuss the special cas
e of cKP - truncated KP hierarchies, obtained as Darboux-Backlund orbi
ts of initial purely differential Lax operators. Our construction esta
blishes the condition for commutativity of the additional-symmetry flo
ws with the discrete Darboux-Backlund transformations of cKP hierarchi
es leading to a new derivation of the string-equation constraint in ma
trix models. (C) 1997 Published by Elsevier Science B.V.