A system of particles experiences a nonzero external force. At an...

According to Newton's second law for a system of particles, the acceleration of the centre of mass is given by $\vec{a}_{cm} = \dfrac{\vec{F}_{ext}}{M}$.

Since the system experiences a nonzero external force ($\vec{F}_{ext} \neq 0$), the acceleration of the centre of mass must be nonzero. Thus, $a_0 \neq 0$.

The velocity of the centre of mass, $v_0$, depends on the initial conditions and the duration for which the force has been applied. It can be either zero (for example, if the system starts from rest or is momentarily at rest) or nonzero.

Therefore, the possible cases are $v_0 = 0, a_0 \neq 0$ and $v_0 \neq 0, a_0 \neq 0$.