On free Gelfand–Dorfman–Novikov–Poisson algebras and a PBW theorem

In 1997, X. Xu [18,19] invented a concept of Novikov–Poisson algebras (we call them Gelfand–Dorfman–Novikov–Poisson (GDN–Poisson) algebras). We construct a linear basis of a free GDN–Poisson algebra. We define a notion of a special GDN–Poisson admissible algebra, based on X. Xu's definition and an S.I. Gelfand's observation (see [9]). It is a differential algebra with two commutative associative products and some extra identities. We prove that any GDN–Poisson algebra is embeddable into its universal enveloping special GDN–Poisson admissible algebra. Also we prove that any GDN–Poisson algebra with the identity x∘(y⋅z)=(x∘y)⋅z+(x∘z)⋅y is isomorphic to a commutative associative differential algebra.