Phase portraits of the full symmetric Toda systems on rank-$2$ groups

We continue investigations begun in our previous works where we proved that the phase diagram of the Toda system on special linear groups can be identified with the Bruhat order on the symmetric group if all eigenvalues of the Lax matrix are distinct or with the Bruhat order on permutations of a multiset if there are multiple eigenvalues. We show that the phase portrait of the Toda system and the Hasse diagram of the Bruhat order coincide in the case of an arbitrary simple Lie group of rank $2$. For this, we verify this property for the two remaining rank-$2$ groups, $Sp(4,\mathbb R)$ and the real form of $G_2$.