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Finiteness of fixed points of a Lie group action

Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$. Under which assumptions there will be a ...
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Questions (doubts) on: Group Action on Manifolds

There are 2 questions that are bugging me in differential topology and I'd be glad if the same could be cleared up: Let $X = x\frac{\partial}{\partial y}$ be a vector field on $M = R^2$, where $R$ ...