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In the paper "A survey of group actions on 4-manifolds" by Allan L. Edmonds on page 5 there is the remark "One should note that the coned-off E8-plumbing manifold admits a circle action that does have more than four orbit types in a neighbourhood of the cone point."

I guess this answers my previous question, but I am not sure since I guess it is possible that the fixed point set it not a submanifold (I only know how to prove this for smooth $S^1$-actions). Could somebody explain how this circle action is constructed? what is the fixed point set? is it a submanifold?

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    $\begingroup$ The coned-off E8 manifold is not a manifold. It has a singular point whose link is the Poincaré homology sphere. $\endgroup$
    – Danny Ruberman
    Commented Nov 2, 2021 at 17:40

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