From the article: “ To see that all these equations of motion are physically possible solutions, it's helpful to use the time reversibility of Newtonian mechanics. It is possible to roll a ball up the dome in such a way that it reaches the apex in finite time and with zero energy, and stops there. By time-reversal, it is a valid solution for the ball to rest at the top for a while and then roll down in any one direction. However, the same argument applied to the usual kinds of domes (e.g., a hemisphere) fails, because a ball launched with just the right energy to reach the top and stay there would actually take infinite time to do so.”
I definitely got that part, I was thinking about just usually a linear ramp to get to the top in a certain amount of time. But I didn't think about the fact that the ball kind of has to rotate around at the top in order to be perfectly balanced so now I see why there is some nuance.
The movement of a ball on each shape (triangular, parabolic, Norton's-domic) will be modeled with its own differential equation. Some of these differential equations will have one and only one solution (given the boundary conditions).
One of these differential equations has an infinite number of solutions--even given the boundary conditions :-)
Best way to really understand this is to roll up your sleeves and spend an afternoon setting up and solving the various differential equations.