  It is well-known that the algebra of vector fields on the circle acts on the
space of Riemann surfaces with a marked point and a local parameter at this
point. We show that this action has a natural realization in the soliton
theory, indeed it coincides with the action of some non-isospectral
Kadomtsev-Petviashvili symmetries on the finite-gap solutions. A technique
based on the so-called Cauchy-Baker-Akhiezer kernel is developed. The
deformations of the \tau-function corresponding to the Baker-Akhiezer forms of
tensor weight j generate representations of the Virasoro algebra with a central
charge 6j^2-6j+1. A system including the Kadomtsev-Petviashvili hierarchy and
the Toda lattice simultaneously is considered. The Virasoro representations
corresponding to such a system explicitly depend on an extra discrete time t_0.
The tau-function for this system is defined in terms of infinite dimensional
flag spaces, generalizing the grassmanians.
