Runaway electron current generated during the Current Quench phase of tokamak disruptions could result in severe damage to future high-performance devices. To control and mitigate such runaway electron current, it is important to accurately describe the runaway electron current dominated equilibrium first, based on which further stability analysis could be carried out. In this study, we show the current status of a Grad-Shafranov-like equation solving for the axisymmetric drift surfaces of the runaway electrons instead of the magnetic flux surfaces for the simple case that all runaway electron shares the same parallel momentum and is the dominant current carrier. This new equilibrium equation is then numerically solved with simple rectangular wall with ITER-like and MAST-like geometry parameters. The deviation between the drift surfaces and the flux surfaces is readily obtained, and runaway electrons is found to be well confined even in regions with open field lines. The change of the runaway electron parallel momentum is found to result in a horizontal current center displacement without any changes in the total current or the external field. The runaway current density profile is found to affect the susceptibility of such displacement, with flatter profiles result in more displacement by the same momentum change. With up-down asymmetry in the external poloidal field, such displacement is accompanied by a vertical displacement of runaway electron current. It is found that this effect is more pronounced in smaller, compact device and weaker poloidal field cases. The above results demonstrate the dynamics of current center displacement caused by the momentum space change in the runaway electrons, and pave way for future, more sophisticated runaway current equilibrium theory with more realistic consideration on the runaway electron momentum distribution. This new equilibrium theory also provides foundation for future stability analysis of the runaway electron current. The next step of the equilibrium theory development would be to move away from given constant parallel momentum and attempt to include finite phase space distribution of the runaways.