Embeddings determined by universal words in the rank $2$ free group

We consider a specific method for embedding a countable group that is given by generators and relations into some $2$-generated group. This embedding enables us to express the images of generators of the countable group in the $2$-generated group and explicitly deduce from the defining relations of the latter those of the former which inherit some special properties. The method can be used to construct the explicit embeddings of recursive groups into finitely presented groups.