Poisson spectra in polynomial algebras

A significant class of Poisson brackets on the polynomial algebra $\text{C}[x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ is studied and, for this class of Poisson brackets, the Poisson prime ideals and Poisson primitive ideals are determined. Moreover it is established that these Poisson algebras satisfy the Poisson Dixmier-Moeglin equivalence.