We study whether the center of relativistic energy (CRE) is always constant in an isolated frame of reference (IFR). First, we assume that two objects in a moving isolated frame of reference (MIFR) move parallel to and very close to a coordinate axis from opposite directions and with equal speed. Here each length in the direction of travel of the objects is much longer than that perpendicular to the direction of travel of them, and moreover the latter is negligibly extremely short. Hence, the location of the CRE (LCRE) of the objects is very close to the coordinate axis in initial condition. When they become perfectly symmetric with respect to the coordinate axis, the forces perpendicular to the direction of travel of the objects are applied to each CRE and thereby a perfectly inelastic collision between them occurs on the coordinate axis. The combined object (CO) resulting from the perfectly inelastic collision begins to rotate because each momentum of the objects before the combination between them acts like the moment of force on the CO. For simplicity, we examine the energy distribution of the CO when it becomes perpendicular to the coordinate axis due rotation. The magnitude of velocity of each minute portion (MP) symmetrical with respect to the coordinate axis are different depending on whether the direction of each rotational velocity is the same or opposite to the direction of travel of the MIFR. Then the energy of each MP is not the same. As a result, the LCRE of the CO moves to the direction perpendicular to the coordinate axis, and its LCRE becomes far from the coordinate axis since the length in such direction is long enough. Therefore, we conclude that the LCRE in the MIFR is not necessarily invariant. Second, we suppose a process in which a force causes a negative acceleration on a moving object (MO) and thereafter the MO eventually comes to rest. What is considered here is the LCRE when the MO comes to rest. We pay attention to the position of the force applied to the MO because the transmission time of force (TTF) inside the MO may vary depending on the position of the MO where the force is applied. As a result, we find that the position at which the MO comes to rest differs depending on the TTF inside the MO. Therefore, we conclude that the LCRE in an IFR differs depending on the position of the MO where the force is applied, in other words, it in an IFR is not necessarily uniquely determined.