form of the field equations [@ll; @fo; @cha1; @cha2; @wei; @mtw]. Our main problem concerns [*the form*]{} of these conservation laws. More precisely, we investigate whether they can be entirely expressed in terms of the basic quantities introduced in the DSX post-Newtonian theory: namely, the set of the relativistic individual mass and spin multipole moments of the $N$ bodies $(M_L^A,S_L^A)$ and the set of the gravitoelectric and gravitomagnetic tidal moments experienced by each body $(G_L^A,H_L^A)$. \[Here, as in Refs. [@dam1; @dam2; @dam3; @dam4] whose notation we follow, $A,B = 1, ..., N$ labels the various bodies, and $L = i_1 ... i_l$ is a multi-spatial index\] The former fully characterize the structure of the post-Newtonian gravitational field generated by an extended body in its local coordinate system, while the latter characterize the tidal action of the other bodies in this local coordinate system, including inertial contributions due to its acceleration and rotation. Obviously, the positions and velocities, with respect to the global coordinate system, of the origins of the local systems attached to each body, as well as their orientations, need to be considered, and will also enter the final expressions.

Let us start by discussing the various methods used 