On one class of exact Poisson structures

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