Suppose I have a function $f$: $\mathbb{R}^n \rightarrow \mathbb{R}$. I'm interested in the symmetry group of $f$, i.e. transformations $T$ such that $f(T(x)) = f(x)$ for all $x$, with composition as the group operation. (Side question: is there a standard name for this group?)
Assuming $f$ is smooth and has compact and connected level sets, the group should also be smooth and compact and connected, so presumably it can be transformed and decomposed into the classical lie groups and/or the exceptional groups. How can I find the decomposition, and especially a mapping from the coordinates of the problem to the usual parameterization(s) of the relevant groups?
For instance, if $f(x) = (x_1^3 + x_1)^2 + x_2^2$ on $\mathbb{R}^2$, then I'd like some procedure which says "this has symmetry isomorphic to $O(2)$", and spits out the action $x' = T(\theta)(x)$ satisfying
$ \begin{bmatrix} (x_1')^3 + x_1' \\ x_2' \end{bmatrix} = R(\theta) \begin{bmatrix} x_1^3 + x_1 \\ x_2 \end{bmatrix} $
where $R(\theta)$ is the $\theta$-rotation matrix (ignoring the discrete symmetry). For quadratic $f$, this is easy, since $T$ is linear, but I want to find the action for more complicated functions $f$.
(If the answer is "go pick up a book on Lie Theory", then I'd really like to know which parts are relevant, especially any particular jargon I should look for.)