We study different algebraic and geometric properties of Heisenberg invariant
Poisson polynomial quadratic algebras. We show that these algebras are
unimodular. The elliptic Sklyanin-Odesskii-Feigin Poisson algebras
$q_{n,k}(\mathcal E)$ are the main important example. We classify all quadratic
$H-$invariant Poisson tensors on ${\mathbb C}^n$ with $n\leq 6$ and show that
for $n\leq 5$ they coincide with the elliptic Sklyanin-Odesskii-Feigin Poisson
algebras or with their certain degenerations.