The G-equivariant Kazdan--Warner problem

We establish an equivariant analogue of the Kazdan--Warner trichotomy for admissible scalar curvature functions. Let $M$ be a closed connected manifold of dimension $n \ge 3$ equipped with an effective isometric action of a compact connected Lie group $G$ of cohomogeneity at least one and with no zero-dimensional orbits. All metrics and prescribed functions are taken to be $G$-invariant. We prove that, for such pairs $(M, G)$, the classical trichotomy does not extend verbatim. A distinct class emerges, consisting of \emph{totally $G$-positive} pairs, for which every $G$-invariant metric exhibits positive total scalar curvature. Each such pair admits a $G$-invariant metric of positive constant scalar curvature, but admits no metric of zero or negative constant scalar curvature. Every remaining pair falls into exactly one of three classes mirroring the original trichotomy.