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