# G-invariant Kazdan--Warner problem

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