The thesis is organized in three distinct parts. The first part ( § 1 and §2) is purely
expository. After a brief introduction to the basic ideas of the bi-Hamiltonian approach
to integrable systems in §1, in §2 the bi-Hamiltonian factorization of Sato's equations is
described as explicitly as possible. This section collects the "experimental facts" which
we aim to explain in the following part of the thesis. In §2 we also introduce and discuss
different representations of the Sato hierarchy, as families of integro-differential equations
in two space variables and in a finite number of fields, which we call Sato-Gel'fand-Dikii
hierarchies. The simplest representation of this type coincides with the well-known KP
hierarchy, while the other representations do not appear, to oμr knowledge, in the previous
literature. The second part of the thesis includes §3, §4 and §5 and represents the theoretical core
of this work. The main result (§4) is the construction of the Poisson-Nijenhuis structures
already mentioned; in §5 we introduce the Kac-Moody algebra of Hamiltonian vector fields,
and we show that these vector fields admit a Lax representation. The only arbitrary point
in the construction is the choice of a Lie-algebra cocycle corresponding to the affine part
of the Lie-Poisson brackets: both the Kac-Moody algebra of bi-Hamiltonian vector fields
and its Lax representation are completely determined by that cocycle.
The third and final part is devoted to the formal application of the abstract framework
to algebras of pseudodifferential operators. In §6 the Gel'fand-Dikii and the Sato-Gel'fandDikii
hierarchies are obtained as reductions (on different affine subspaces) of the dynamical
systems previously obtained, for finite n, while the Sato hierarchy is recovered in §7 as a
generalization of the same construction to the case n = oo.
