Newtonian gravitational multipoles as group-invariant solutions

A family of vector fields that are the infinitesimal generators of determined one-parameter groups of transformations are constructed. It is shown that these vector fields represent symmetries of the system of differential equations interrelated by the axially symmetric Laplace equation and a certain supplementary equation. Group-invariant solutions of this system of equations are obtained by means of two alternative methods, and it is proved that these solutions turn out to be the family of axisymmetric potentials related to specific gravitational multipoles. The existence of these symmetries provides us with a generalization of the fact that the Newtonian monopole is defined by the solution of the Laplace equation with spherical symmetry, and it allows us to extract from all solutions of this equation those with the prescribed Newtonian multipole moments.