Let's work over the complex numbers. Let $S$ be a normal surface, $\mathrm{A}^1(S)$ the class group of divisors on $S$ and $\mathrm{Pic}(S)$ its Picard group. Let $G$ be a reductive group acting on $S$.

Q1. When is $\mathrm{A}^1(S)$ generated by $G$-invariant divisors?

Q2. Assume $S$ smooth. When is $\mathrm{Pic}(S)$ generated by (line bundles associated to) $G$-invariant divisors?

Q3. What if $G$ is the multiplicative group $\mathbb{C}^*$?

**Remark:** I'm looking for answers in which $S$ is *not* toric with torus $G$, because I consider that case already known.