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.