Let $T=S^{1}\times S^{1}\times ...\times S^{1}$ ($n$ times) be $n$ dimensional torus and $X$ be a $T$-space.

Lemma: If $X$ has finitely many connected orbit type, then there is a subcircle $L=S^{1}\subseteq T$ such that $X^{L}=X^{T}$ (fixed point sets of the action).

I want to prove this lemma.

In some books, The hypotesis finitely many connected orbit type is not necessary. That is, There is always a subcircle $L=S^{1}\subseteq T$ such that $X^{L}=X^{T}$. Is this true?

If $G$ is a compact connected Lie group, then there is a subcircle $% L=S^{1}\subseteq G$ such that $X^{L}=X^{G}$? That is, there is subtorus $T\subseteq G$ such that $X^{T}=X^{G}$?