On the special identities of Gelfand--Dorfman algebras

In this paper, we prove that the class of all special Gelfand--Dorfman algebras (GD-algebras) is closed with respect to homomorphisms and thus forms a variety. We also prove that every 2-dimensional GD-algebra is special. For the latter, we give a technical method to find all special identities of GD-algebras and compute the degree 6 component of the Gr\"obner basis for the shuffle operad constructed on the symmetric operad governing the class of GD-algebras.