  The gauge equivalence between basic KP hierarchies is discussed. The first
two Hamiltonian structures for KP hierarchies leading to the linear and
non-linear $\Winf$ algebras are derived. The realization of the corresponding
generators in terms of two boson currents is presented and it is shown to be
related to many integrable models which are bi-Hamiltonian. We can also realize
those generators by adding extra currents, coupled in a particular way,
allowing for instance a description of multi-layered Benney equations or
multi-component non-linear Schroedinger equation. In this case we can have a
second Hamiltonian bracket structure which violates Jacobi identity. We
consider the reduction to one-boson systems leading to KdV and mKdV
hierarchies. A Miura transformation relating these two hierarchies is obtained
by restricting gauge transformation between corresponding two-boson
hierarchies. Connection to Drinfeld-Sokolov approach is also discussed in the
$SL(2,\IR)$ gauge theory. (Lectures presented at the VII J.A. Swieca Summer
School, Section: Particles and Fields, Campos do Jord\~ao - Brasil -
January/93)
