  A method to construct interior axially symmetric metrics that appropriately
match with any vacuum solution of the Weyl family is developed in
Hernandez-Pastora etal. (Class Quantum Gravity 33:235005, 2016). It was
shown,for the case of some vacuum solutions, that the simplestsolution for the
interior metric leads to sources with well behaved energy conditions. Now, we
integrate the field equa-tions to obtain the interior metric functions in terms
of theanisotropies and pressures of the source. As well, the compatible
equations of state for these global models are calculated. The interior metric
and the suitable energy momentum tensor describing the source are constructed
in terms of the exterior metric functions. At the boundary of the compact
object,the behaviour of a pressure Tm, defined from the energy momentum tensor,
is shown to be related with the exterior gravitational field. This fact allows
us to explore the differences arising at the matter distribution when the
sphericalsymmetry of the global metric is dropped. Finally, an equation derived
from the matching conditions is obtained whichallows us to calculate the Weyl
coefficients of the exteriormetric as source integrals. Hence the Relativistic
MultipoleMoments of the global model can be expressed in terms ofthe matter
distribution of the source.
