  We show that manifolds which parameterize values of first integrals of
integrable finite-dimensional bihamiltonian systems carry a geometric structure
which we call a {\em Kronecker web}. We describe two functors between Kronecker
webs and integrable bihamiltonian structures, one is left inverse to another
one. Conjecturally, these two functors are mutually inverse (for ``small'' open
subsets).
  The above conjecture is proven provided the bihamiltonian structure allows an
antiinvolution of a particular form. This implies the conjecture of
\cite{GelZakh99Web} that on a dense open subset the bihamiltonian structure on
${\mathfrak g}^{*}$ is flat if ${\mathfrak g}$ is semisimple, or if ${\mathfrak
g}={\mathfrak G}\ltimes \operatorname{ad}_{{\mathfrak G}}$ and ${\mathfrak G}$
is semisimple, and for some other Lie algebras of mappings.
