  We introduce a criterion that a given bihamiltonian structure allows a local
coordinate system where both brackets have constant coefficients. This
criterion is applied to the bihamiltonian open Toda lattice in a generic point,
which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of
brackets with constant coefficients. This shows that the open Toda lattice
cannot be locally represented as a product of two bihamiltonian structures.
  In a generic point the bihamiltonian periodic Toda lattice is shown to be
isomorphic to a product of two open Toda lattices (one of which is a (trivial)
structure of dimension 1).
  While the above results might be obtained by more traditional methods, we use
an approach based on general results on geometry of webs. This demonstrates a
possibility to apply a geometric language to problems on bihamiltonian
integrable systems, such a possibility may be no less important than the
particular results proven in this paper.
  Based on these geometric approaches, we conjecture that decompositions
similar to the decomposition of the periodic Toda lattice exist in local
geometry of the Volterra system, the complete Toda lattice, the
multidimensional Euler top, and a regular bihamiltonian Lie coalgebra. We also
state general conjectures about geometry of more general ``homogeneous''
finite-dimensional bihamiltonian structures.
  The class of homogeneous structures is shown to coincide with the class of
system integrable by Lenard scheme. The bihamiltonian structures which allow a
non-degenerate Lax structure are shown to be locally isomorphic to the open
Toda lattice.
