  K\"ahler-Poisson algebras were introduced as algebraic analogues of function
algebras on K\"ahler manifolds, and it turns out that one can develop geometry
for these algebras in a purely algebraic way. A K\"ahler-Poisson algebra
consists of a Poisson algebra together with the choice of a metric structure,
and a natural question arises: For a given Poisson algebra, how many different
metric structures are there, such that the resulting K\"ahler-Poisson algebras
are non-isomorphic? In this paper we initiate a study of such moduli spaces of
K\"ahler-Poisson algebras defined over rational functions in two variables.
