  We prove that the Poisson version of the Dixmier-Moeglin equivalence holds
for cocommutative affine Poisson-Hopf algebras. This is a first step towards
understanding the symplectic foliation and the representation theory of
(cocommutative) affine Poisson-Hopf algebras. Our proof makes substantial use
of the model theory of fields equipped with finitely many possibly noncommuting
derivations. As an application, we show that the symmetric algebra of a finite
dimensional Lie algebra, equipped with its natural Poisson structure, satisfies
the Poisson Dixmier-Moeglin equivalence.
