I have a surface defined by the implicit formula: $$ F(\boldsymbol{x}) = 0 $$ where $ \boldsymbol{x} \in \mathbb{R}^n$ and $F : \mathbb{R}^n \to \mathbb{R} $ (actually I have $n=3$, but if the result is in a generic $\mathbb{R}^n$ it would be nice). Let's suppose to have $\nabla F(\boldsymbol{x})$, the gradient of $F$. Let's define the vector function $$ \boldsymbol{\boldsymbol{G}}(\boldsymbol{x}) = \nabla F(\boldsymbol{x}) $$ which maps $\mathbb{R}^n$ in $\mathbb{R}^n$. Now, suppose that $\boldsymbol{G}$ is invertible and we can compute $\boldsymbol{G}^{-1}(\boldsymbol{x})$.
I'm facing the following problem: find a scalar $\lambda>0$ such that $$ F\left({\boldsymbol{G}^{-1}\left(\frac{1}{\lambda}\boldsymbol{x}\right)}\right) = 0. $$ My question is if any existing result about gradients can help me simplify the formula and get closer to writing something like $\lambda = f(\boldsymbol{x})$. If I need additional hypotheses on the surface properties it is fine.
For example, if the surface is an ellipsoid and $F(\boldsymbol{x}) = \boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - 1$, with $\boldsymbol{A}\in\mathbb{R}^{3\times 3}$ a diagonal positive definite matrix, we have $\lambda = \frac{1}{2}\sqrt{\boldsymbol{x}^T\boldsymbol{A}^{-1}\boldsymbol{x}}$. Is this result generalizable?