I asked the following question over at [math.stackexchange](https://math.stackexchange.com/questions/191032/when-is-an-orbit-spherical), but got no answers. Maybe it's less well-known than I thought, but I still wanted to ask here:

Let's assume we have an affine, reductive, algebraic group $G$ acting algebraically on a variety $X$, everything over an algebraically closed field of characteristic zero. Let $x\in X$ be some point with reductive stabilizer $H:=G_x$. Under what conditions on $x$ or $H$ is the orbit $G.x\cong G\newcommand{\qq}{/\hspace{-.8ex}/}\qq H$ a *spherical variety*? Let me briefly recall that a spherical variety is a homogeneous space $G\qq H$ satisfying one of the following, equivalent properties:

 1. Any Borel subgroup $B\subseteq G$ has an open orbit in $G\qq H$.
 2. Every equivariant completion of $G\qq H$ contains only finitely many orbits.
 3. For every irreducible $G$-module $V$ and any character $\chi$ of $H$,
   $$\dim\left\{~v\in V \mid \forall h\in H: h.v = \chi(h)v ~\right\}\le 1.$$

I was hoping that this is well-known, but I cannot find any direct statements of that kind. Searching for the keywords "orbit" and "spherical" is quite fruitless because of property 1. 

<b>Edit:</b> In the cases of interest to me, the orbit $G.x$ is affine.