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The symmetry algebra $P_\infty = W_\infty \oplus H \oplus I_\infty$ ofintegrable systems is defined. As an example the classical Sophus Lie pointsymmetries 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 partof the symmetries. The negative part can not be obtained from the free fermionalgebra. A new embedding of the Virasoro algebra into $gl(\infty )$n describesconformal transformations of the KP time variables. A free fermion algebracocycle is described as a PDO Lie algebra cocycle.Comment: 21 pages, Latex, no figures (some references added and misprints are  corrected

P∞ Algebra of KP, Free Fermions and 2-Cocycle in the Lie Algebra of Pseudodifferential Operators

P_∞ Algebra of KP, Free Fermions and 2-Cocycle in the Lie Algebra of Pseudodifferential Operators