  We study polynomial Poisson algebras with some regularity conditions. Linear
(Lie-Berezin-Kirillov) structures on dual spaces of semi-simple Lie algebras,
quadratic Sklyanin elliptic algebras of \cite{FO1},\cite{FO2} as well as
polynomial algebras recently described by Bondal-Dubrovin-Ugaglia
(\cite{Bondal},\cite{Ug}) belong to this class. We establish some simple
determinantal relations between the brackets and Casimirs in this algebras.
These relations imply in particular that for Sklyanin elliptic algebras the sum
of Casimir degrees coincides with the dimension of the algebra. We are
discussing some interesting examples of these algebras and in particular we
show that some of them arise naturally in Hamiltonian integrable systems. Among
these examples is a new class of two-body integrable systems admitting an
elliptic dependence both on coordinates and momenta.
