This talk will concentrate on the local geometry only, and only near points in "general position." We describe how to recognize such points, and which geometric objects one can construct near such points. It turns out that some of these objects are defined canonically, some only up to a diffeomorphism.
For example, near points in general position any bihamiltonian structure may be decomposed into a product of two parts, an "odd" one and an "even" one. Such a decomposition is not unique, but the factor bihamiltonian structures are defined uniquely up to a diffeomorphism.
Investigation of the associated (simpler) geometric structures leads to a way of locally classifying bihamiltonian structures in general position. This provides an enormous pool of non-isomorphic "odd" bihamiltonian structures, each leading to a different integrable system.
This classification also provides an algorithm to check whether two
bihamiltonian structures are isomorphic to each other. Surprisingly,
the "classical" integrable systems turn out to be isomorphic to a product
of several open Toda lattices. This reduces the differences among
these systems to be either of a global nature, or in the geometry of the
"discriminant set," where the bihamiltonian structure is not in general
position.
1. Associate to a Poisson bracket { , } the collection (=pseudofoliation) 
of the symplectic leaves of { , }; associate to a bihamiltonian structure 
1{
, }1 + 2{
, }2 the family 
of the corresponding pseudofoliations. If we know the bihamiltonian structure
up to diffeomorphism, then we know the family
up to diffeomorphism.
Say that the bihamiltonian structure is webby if the opposite
is also true: if another bihamiltonian structure has the same pseudofoliations ,
then these two structure are diffeomorphic.
There is a strong evidence that any bihamiltonian structure is locally webby on a dense open subset. (Note that the analogue of this property holds for Poisson structures---on the subset where the structure has locally constant rank. However, it is not interesting, since the only invariant of both a foliation and a Poisson structure is the rank.) Is this true? Is this true for some larger class of Poisson structures depending on a parameter?
2. Consider a family { , }s of Poisson structures
on a manifold M which depends on a parameter s 
S;
here S is an algebraic variety. Is there a natural construction
which associates to (M,{ , }s) another pair 
,
such that 
is a bihamiltonian
structure? (In other words,
depends linearly on .)
3. A bihamiltonian structure is locally flat if it is locally isomorphic to a bihamiltonian structure on Cd with structure tensors having constant coefficients. There is a strong evidence that such structures should be related to Lax representations of integrable systems. Call a bihamiltonian structure strongly integrable if it is locally flat on a dense open subset.
Is it true that the dynamic of strongly integrable bihamiltonian structures allows Lax representations? Is there an inverse relationship?
4. Consider a complex of differential operators
-manifold
M.
Say that this complex is of principal type if the sheaf of cohomology
of this complex ``changes by finite-dimensional terms only'' when we vary
the operators 
,
but the principal symbols remain the same during the variation (and the
property k
ok-1
=
0 holds!) Is it possible to describe all such complexes?
One example of such a complex is the de Rham complex.