  We perform a comprehensive analysis of the set of parameters $\{r_{i}\}$ that
provide the energy distribution of pure qutrits that evolve towards a
distinguishable state at a finite time $\tau$, when evolving under an arbitrary
and time-independent Hamiltonian. The orthogonality condition is exactly
solved, revealing a non-trivial interrelation between $\tau$ and the energy
spectrum and allowing the classification of $\{r_{i}\}$ into families organized
in a 2-simplex, $\delta^{2}$. Furthermore, the states determined by $\{r_{i}\}$
are likewise analyzed according to their quantum-speed limit. Namely, we
construct a map that distinguishes those $r_{i}$s in $\delta^{2}$ correspondent
to states whose orthogonality time is limited by the Mandelstam--Tamm bound
from those restricted by the Margolus--Levitin one. Our results offer a
complete characterization of the physical quantities that become relevant in
both the preparation and study of the dynamics of three-level states evolving
towards orthogonality.
