Divergent Trajectories in Arithmetic Homogeneous Spaces of Rational Rank Two

Let $G$ be a real algebraic group defined over $\mathbb{Q}$, $\mathrm{\Gamma }$ be an arithmetic subgroup of $G$, and $T$ be a maximal $\mathbb{R}$-split torus. A trajectory in $G\mathrm{/}\mathrm{\Gamma }$ is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which account for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in $T$, we study the existence of non-obvious divergent trajectories under its action in $G\mathrm{/}\mathrm{\Gamma }$. We get a sufficient condition for the existence of a non-obvious divergence trajectory in the general case, and a full classification under the assumption that $ran{k}_{\mathbb{Q}}G=ran{k}_{\mathbb{R}}G=2$.