I am trying to solve the following equation: $$ F(\mathbf{a}) = G(\mathbf{b}) $$ where $F,G: \mathbb{R}^n \rightarrow \mathbb{R}$. The expressions for $F$ and $G$ are quite complicated, and I'm wondering if group theory would offer any advantage in solving this equation (i.e. finding the relationship between $\mathbf{a}$ and $\mathbf{b}$).
For example if I am able to find all linear transformations $X \in GL(n,\mathbb{R})$ such that $$ F(X \mathbf{a}) = F(\mathbf{a}) $$ and similarly for $G$, would that offer any insight into the solution of this problem? By solution I would like to find some relationship between $\mathbf{a}$ and $\mathbf{b}$.
Just wondering if group theory has ever been applied to this sort of problem, and if there are any books/references for this I'd appreciate it.