Lectures on classical W-algebras

These are lecture notes of lectures given in 1993 in Cortona, Italy. W-algebras appeared in the conformal field theory as extensions of the Virasoro algebra. They are closely connected with integrable systems and can be interpreted as symplectic (or Poisson) structures inherent in those systems (we touch only classical aspects of W-algeabras). We try to give some self-contained description of this link. First three parts are devoted to integrable systems and their Hamiltonian nature. Part 4 deals with the most important for W-algebras problem: seeking generators that are primary fields. Parts 4 and 5 introduce the KP-hierarchy and tau-functions, Part 6 discusses additional symmetries, W-infinity-algebra and the string equation. Finally, Part 7 is devoted to some reductions (constrained hierarchies).