Spherical homogeneous spaces of minimal rank

Let $G$ be a complex connected reductive algebraic group and $G\mathrm{/}B$ denote the flag variety of $G$. A $G$-homogeneous space $G\mathrm{/}H$ is said to be {\it spherical} if $H$ acts on $G\mathrm{/}B$ with finitely many orbits. A class of spherical homogeneous spaces containing the tori, the complete homogeneous spaces and the group $G$ (viewed as a $G\times G$-homogeneous space) has particularly nice proterties. Namely, the pair $(G,H)$ is called a {\it spherical pair of minimal rank} if there exists $x$ in $G\mathrm{/}B$ such that the orbit $H\mathrm{.}x$ of $x$ by $H$ is open in $G\mathrm{/}B$ and the stabilizer $H_x$ of $x$ in $H$ contains a maximal torus of $H$. In this article, we study and classify the spherical pairs of minimal rank.