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The monopole solution of the Einstein vacuum field equations (Schwarzschild`ssolution) in Weyl coordinates involves a metric function that can beinterpreted as the gravitational potential of a bar of length $2m$ withconstant linear density. The question addressed in this work is whether similarrepresentations can be constructed for Weyl solutions other than thespherically symmetric one.  A new family of static solutions of the axisymmetric vacuum field equationsgeneralizing the M-Q$^{(1)}$ solution is developed. These represent slightdeviations from spherical symmetry in terms of the relativistic multipolemoments (RMM) we wish the solution to contain. A Newtonian object referred toas a dumbbell can be used to describe these solutions in a simple form by meansof the density of this object, since the physical properties of therelativistic solution are characterized by its behaviour. The density profileof the dumbbell, which is given in terms of the RMM of the solution, allows usto distinguish general multipole Weyl solutions from the constant-densitySchwarzschild solution. The range of values of the multipole moments thatgenerate positive-definite density profiles are also calculated. The bounds onthe multipole moments that arise from this density condition are identical tothose required for a well-behaved infinite-redshift surface $g_{00}=0$.

Linearized multipole solutions and their representation