Let $x\in M$, $M$  finite dimensional smooth manifold. Is there an example of a finite dimensional Lie group action on $M$ with no slice at $x$?
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Recall that if a free action of G on M has a slice S at a point x then the natural map of G x S into M given by (g,s) maps to gs would be a diffeomorphism onto a tubular neighborhood of the orbit Gx. So for a counterexample take the action of the real line on the 2torus given by a 1parameter subgroup with irrational slope acting by translation. 

