### 5.1 Super-potential method

Up to 2.5 PN order, we have solved all the Poisson equations necessary to derive the 2.5 PN gravitational
field. At 3 PN order, we have found a part of the solutions of the Poisson equations, which we call
(super-)potentials.
For example,
where , and which satisfies is given in [99] as
It is possible to add any homogeneous solution to super-potentials. In our formalism, the only place
where we use super-potentials is Equation (101). In the case above, we could add, say, to .
(Note that to evaluate the surface integrals in the general form of equations of motion (111)
we need super-potentials in the spatial region which do not include any singularity
due to the point particle limit.) It is easy to see that contribution from a possible additional
homogeneous solution cancels out between the “” term and the surface integral in
Equation (101).

Useful super-potentials for derivation of the 3 PN field are given in [30, 27, 91, 99].