Commutative Leibniz–Poisson algebras of polynomial growth

In this paper we study commutative Leibniz–Poisson algebras. We prove that a variety of commutative Leibniz–Poisson algebras has either polynomial growth or growth with exponential not less than 2, the field being arbitrary. We prove that every variety of commutative Leibniz–Poisson algebras of polynomial growth over a field of characteristic 0 has a finite basis for its polynomial identities. Also we construct a variety of commutative Leibniz– Poisson algebras with almost polynomial growth.