  The symmetry algebra $P_\infty = W_\infty \oplus H \oplus I_\infty$ of
integrable systems is defined. As an example the classical Sophus Lie point
symmetries of all higher KP equations are obtained. It is shown that one
(``positive'') half of the point symmetries belongs to the $W_\infty$
symmetries while the other (``negative'') part belongs to the $I_\infty$ ones.
The corresponing action on the tau-function is obtained for the positive part
of the symmetries. The negative part can not be obtained from the free fermion
algebra. A new embedding of the Virasoro algebra into $gl(\infty )$n describes
conformal transformations of the KP time variables. A free fermion algebra
cocycle is described as a PDO Lie algebra cocycle.
