The motion of a two-body system under their internal forces (assuming no external forces) can be studied using the Lagrangian analysis.
We assume that both the bodies experience equal and opposite forces and rotate around the common centre of mass. Then, the differential equations describing the orbits (see (1), (2) and (3) in image) have the reduced mass terms to represent combined motion of both the masses.
In these equations, if we assume that the central mass is very large compared to the planet mass, then the planet mass (m2) term vanishes and orbit equations become dependent only on the mass of the central body (m1). This situation is similar to the Schwarzschild solution results (eq. (4) in image) for a large central mass. The GR analysis suggests that the space-time around the large central mass gets curved and therefore the planets have to necessarily follow a curved path. The nature of the curved path is independent of the mass of the planet. Therefore, even a photon (which has zero mass) path has to bend as the space itself is curved.
But, we may also have a situation where the planet mass is not very small compared with the central star. Then according to the classical analysis, the rotation of both the bodies around the common centre will have to be considered. In a classical analysis, this two-body system can be represented mathematically as a single reduced mass rotating around the centre. Similarly, an analysis for the two-body system, based on GR should also consider the resultant curvature created by the both masses.
The question is: How to include the reduced mass term in the GR analysis? This is necessary to ensure that the results of the classical analysis and GR are consistent with each other.