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Let $M$ be a smooth manifold and $G$ a Lie group acting properly on $M$. Let $k$ be the codimension of a maximal dimensional $G$-orbit in $M$.
Is it always possible to construct $k$ functions $f_1, \dots, f_k \colon M \to \mathbb{R}$, such that $f_1, \dots , f_k$ are $G$-invariant and functionally independent on an open and dense subset of $M$?
If it is not possible, under which further assumptions could it be true?
This can be done locally. Suppose that the manifold $M$ and the Lie group $G$ are compact. Then one can say that the type of the orbits of $x$ and $y$ coincide if and only if the stabilizer of $x$ and $y$ are conjugated. There is particular type called the principal type: The type of the orbit of $x$ is principal if and only if the canonical representation of the stabilizer $G_x$ on $T_xM/V_x$) where $T_xM$ is tangent space of $M$ at $x$ and $V_x$ is the subspace of $T_xM$ tangent to the orbit) is trivial. A principal orbit has a maximal dimension and the set of principal orbit is a dense open subset.
The slice theorem of Koszul allows then to identify equivariantly a neighborhood of the orbit of $x$ to $G/G_x\times W_x$ where is an open subset of $T_xM/V_x$. So you can find functions $f_1,...,f_k$ that are independent functionally and invariant by $G$ given by the coordinates of $W_x$.
