  We investigate the additional symmetries of several supersymmetric KP
hierarchies: the SKP hierarchy of Manin and Radul, the $\hbox{SKP}_2$
hierarchy, and the Jacobian SKP hierarchy. In all three cases we find that the
algebra of symmetries is isomorphic to the algebra of superdifferential
operators, or equivalently $\SW_{1+\infty}$. These results seem to suggest that
despite their realization depending on the dynamics, the additional symmetries
are kinematical in nature.
