If a particle and a system have zero momentum

[DJW]

Is it true that they both must also have zero kinetic energy?

 

[qspeechc]

If we are talking of one particle, and it has zero momentum, it must have zero velocity (mass cannot be zero), and so its kinetic energy is zero.
If a system of particles has zero momentum, anyone particle in the system may still have momentum, say ,if it is vibrating in a lattice, but the system as a whole does not.