  We report a detailed study of the stationary points (zero-force points) of
the potential energy surface (PES) of a model structural glassformer. We
compare stationary points found with two different algorithms (eigenvector
following and square gradient minimization), and show that the mapping between
instantaneous configuration and stationary points defined by those algorithms
is as different as to strongly influence the instability index K vs.
temperature plot, which relevance in analyzing the liquid dynamics is thus
questioned. On the other hand, the plot of K vs. energy is much less sensitive
to the algorithm employed, showing that the energy is the good variable to
discuss geometric properties of the PES. We find new evidence of a geometric
transition between a minima-dominated phase and a saddle-point-dominated one.
We analyze the distances between instantaneous configurations and stationary
points, and find that above the glass transition, the system is closer to
saddle points than to minima.
