This question is in the spirit of Norton's dome, an example of an apparently non-deterministic system in Newtonian mechanics. Under certain restrictions, the Picard–Lindelöf theorem guarantees the existence and uniqueness of solutions to differential equations with initial conditions. However, Norton's dome does not satisfy these restrictions. The ball may sit on top of the dome for an arbitrary period of time before sliding down in a particular trajectory, without violating Newton's laws of motion.
Is there a plausible example of a system of interacting point-particles such that the equations of a point-particle are continuous but not Lipschitz continuous, and therefore fail to have a unique solution?