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We consider a hierarchy of Poisson structures defined on rational functionson the Riemann sphere. This hierarchy is originated in the theory of theintegrable Camassa-Holm equation associated with the Krein's string spectralproblem. Previously the proof of Jacobi identity was obtained by reducing thebracket to canonical Darboux coordinates. The main result of this note is adirect proof of the Jacobi identity. It turns out that the direct proof of theJacobi identity is far from trivial. We also give an example of anotherhierarchy of Poisson brackets and construct Darboux coordinates for it.

Analytic Poisson brackets on rational functions on the Riemann sphere and their generalization