Growth in Poisson algebras

A criterion for polynomial growth of varieties of Poisson algebras is stated in terms of Young diagrams for fields of characteristic zero. We construct a variety of Poisson algebras with almost polynomial growth. It is proved that for the case of a ground field of arbitrary characteristic other than two, there are no varieties of Poisson algebras whose growth would be intermediate between polynomial and exponential. Let $V$ be a variety of Poisson algebras over an arbitrary field whose ideal of identities contains identities
$$ \{\{x_1,y_1\},\{x_2,y_2\},\dots,\{x_m,y_m\}\}=0,\qquad\{x_1,y_1\}\cdot\{x_2,y_2\}\cdot\ldots\cdot\{x_m,y_m\}=0, $$
for some $m$. It is shown that the exponent of $V$ exists and is an integer.
For the case of a ground field of characteristic zero, we give growth estimates for multilinear spaces of a special form in varieties of Poisson algebras. Also equivalent conditions are specified for such spaces to have polynomial growth.