On one class of exact Poisson structures

We discuss some properties of a natural class of Poisson structures on Eucl idean spaces and abstract manifolds. In particular it is proved that such s tructures are always exact and may be reconstructed from their Casimir func tions. It is shown that in low dimensions they give the whole class of exac t Poisson structures. The dimension of Poisson homology of these structures is computed in terms of the Milnor number of their Casimir functions. We a lso analyze some concrete examples of such structures in low dimensions and show that their centers are generated by Casimir functions. (C) 2001 Ameri can Institute of Physics.