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].
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