Surjectivity for systems of equations over groups

Determine whether the natural map from a coefficient group \(G\) to a group defined by a balanced system of more than one equations over \(G\) is surjective, beyond the cases covered by the known results cited in the paper.

Background

The paper relates surjectivity of the substitution homomorphism $\epsilon:G\to\hat{G}$ to adjunction problems and systems of equations over groups. It notes that surjectivity requires both a trivial substitution group and normality of the image $\epsilon(G)$ in $\hat{G}$.

The cited literature establishes substantial results for one equation, including restrictions in the torsion-free case, but the authors report that the corresponding problem for systems of equations remains unresolved.