The center of momentum of a system is not a location, but a collection of relative momenta/velocities: a reference frame.
In all center-of-momentum frames, the center of mass is at rest, but it is not necessarily at the origin of the coordinate system.
In special relativity, the COM frame is necessarily unique only when the system is isolated.
Its energy is – for each reference frame – equal to the magnitude of momentum multiplied by the speed of light: An example of the usage of this frame is given below – in a two-body collision, not necessarily elastic (where kinetic energy is conserved).
The situation is analyzed using Galilean transformations and conservation of momentum (for generality, rather than kinetic energies alone), for two particles of mass m1 and m2, moving at initial velocities (before collision) u1 and u2 respectively.