  Let $G$ be a reductive group over an algebraically closed subfield $k$ of
$\mathbb{C}$ of characteristic zero, $H \subseteq G$ an observable subgroup
normalized by a maximal torus of $G$ and $X$ an affine $k$-variety acted on by
$G$. Popov and Pommerening conjectured in the late 70's that the invariant
algebra $k[X]^H$ is finitely generated. We prove the conjecture for 1)
subgroups of $\mathrm{SL}_n(k)$ closed under left (or right) Borel action and
for 2) a class of Borel regular subgroups of classical groups. We give a
partial affirmative answer to the conjecture for general regular subgroups of
$\mathrm{SL}_n(k)$.
