Hmm, unless I'm going crazy here, we get the differential equation x'' = -(x^-2) x = (kt)^(2/3)(with k = (2/9)^(-3/2), not that it matters) seems to be a s

Hmm, unless I'm going crazy here, we get the differential equation x'' = -(x^-2) x = (kt)^(2/3)(with k = (2/9)^(-3/2), not that it matters) seems to be a solution? I guess that doesn't help with the singularity, but neither does looking at energy (since you have infinite kinetic energy at the center and infinite potential energy everywhere else.)