Actually, there is nothing mysterious here. The theory, applied correctly, predicts the expected thing: a particle that starts at rest at the apex will never leave it. The other "solution" is not actually a particle at rest (it has a non-0 6th derivative of position, that is the rate of change of the rate of change of its acceleration is 1/6). So it's actually describing a particle whose acceleration will increase at an increasing rate forever, but that just happens to be 0 at time T.