A reference for an equivariant Morse Lemma

Question

Does anybody knows a reference for the following statement?

Let $S^1$ acts on $\mathbb{C}^n$ in the usual (diagonal) way and $f:\mathbb{C}^n\to\mathbb{R}$ a smooth $S^1$-invariant function defined in a neighborhood of $0$ with a non-degenerate critical point at 0. Then, there exists a local $S^1$-equivariant diffeomorphism $\Phi:U\to V\subset \mathbb{C}^n$ such that

$$f(\Phi(z_1,\ldots,z_n))=|z_1|^2+\cdots+|z_p|^2-|z_{p+1}|^2-\cdots-|z_n|^2$$ It looks like Moser's proof of Morse lemma extends easily to this version, but I would be happy to know a place where it is actually done. Thanks for any help.

Answer 1

Check Wasserman, Arthur G. Equivariant differential topology. Topology 8 1969 127--150. MR0250324 (40 #3563). In particular see Lemma 4.1 for an equivariant Morse lemma.