Let $V=Spec(R[V])$ an affine variey and $G$ a linear reductive group action on $V$. (I'm not sure if the linear reductive condition is neccessary for the question).
According to GIT the "quotient" $V/G=Spec(R[V]^G)$ parametrizes closed orbits.
My question is if it can be happen that $V/G$ has more than one (closed) singleton-orbits. Obviously the "$0$-point" is a one-point orbit. Could there also exist another such one-point orbit?
If the answer is yes, then can $V/G$ be even a discrete or finite set?
I we futhermore add extra assumptions on $V$ from topological nature (irreducibility, reduceness,...). What does $V/G$ inherit?