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Kazdan and Warner at the $70$'s published a serie of papers on the problem of ensuring the existence of metrics with prescribed scalar curvature.

For instance:

https://projecteuclid.org/euclid.bams/1183533899

https://projecteuclid.org/euclid.jdg/1214432678

etc

I am wondering the following: if one assumes that a compact Lie group $G$ acts by isometries on the Riemannian manifold $(M,g)$ and give a $G$-invariant function $K : M \to \mathbb{R}$, the existence of a $G$-invariant metric $\tilde g$ on $M$ with $K$ as its scalar curvature follows immediately from their work? If not, is there any reference in the literature where I can find it?

Thanks

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