  New theorems about the existence of solution for a system of infinite linear
equations with a Vandermonde type matrix of coefficients are proved. Some
examples and applications of these results are shown. In particular, a kind of
these systems is solved and applied in the field of the General Relativity
Theory of Gravitation. The solution of the system is used to construct a
relevant physical representation of certain static and axisymmetric solution of
the Einstein vacuum equations. In addition, a newtonian representation of these
relativistic solutions is recovered. It is shown as well that there exists a
relation between this application and the classical Haussdorff moment problem.
