On the application of the Routh equimomental systems in the problem of modeling the potential of Newtonian gravity

For an arbitrary rigid body, a family of Routh equimomental systems is constructed, determined by seven independent parameters. Each solution from the family corresponds to a system of five point masses: four points form a non-degenerate tetrahedron, and the fifth one is located at its center of mass. The solution found is the broadest generalization of previously obtained results and contains them as particular cases. The solution does not allow further generalization without increasing the number of point masses of system. For a rigid body and its Routh equimomental system, the inertia integrals up to the second order coincide. The non-uniqueness of equal momental systems allows one to pose the problem of finding a system that best approximates the moments of mass distribution of higher orders.