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The FHP algorithm allows to obtain the relativistic multipole moments of avacuum stationary axisymmetric solution in terms of coefficients which appearin the expansion of its Ernst's potential on the symmetry axis. First of all,we will use this result in order to determine, at a certain approximationdegree, the Ernst's potential on the symmetry axis of the metric whose onlymultipole moments are mass and angular momentum.  By using Sibgatullin's method we analyse a series of exacts solutions withthe afore mentioned multipole characteristic. Besides, we present anapproximate solution whose Ernst's potential is introduced as a power series ofa dimensionless parameter. The calculation of its multipole moments allows usto understand the existing differences between both approximations to theproposed pure multipole solution.Comment: 24 pages, plain TeX. To be published in General Relativity and  Gravitatio

Approaches to the Monopole-Dynamic Dipole Vacuum Solution Concerning the Structure of Its Ernst Potential on the Symmetry Axis