The phase-portrait conjecture for generalized full symmetric Toda systems

For a semisimple Lie group, consider the associated generalized full symmetric Toda system and assume that all eigenvalues are distinct. The stationary points are compared with elements of the corresponding Weyl group, and trajectories between stationary points are compared with Bruhat intervals.

Phase-portrait conjecture. The phase picture of the Toda system coincides with the Bruhat order on the corresponding Weyl group: stationary points correspond bijectively to Weyl-group elements, and there is a trajectory between two such points if and only if the corresponding elements are comparable in Bruhat order. The dimension of the space of such trajectories equals the number of points in the corresponding Bruhat interval, and the Morse index of a point equals the length of the corresponding Weyl-group element. When eigenvalues coincide, the analogous statement should hold with the Bruhat order on the corresponding partial flag space.

This conjecture extends the previously established identification of phase portraits with Bruhat diagrams for Toda systems on special linear groups. The paper investigates the claim for rank-two groups, including $Sp(4,\mathbb R)$ and the real form of $G_2$; its general validity for arbitrary semisimple Lie groups is not established here.