  In axial symmetry, there is a gauge for Einstein equations such that the
total mass of the spacetime can be written as a conserved, positive definite,
integral on the spacelike slices. This property is expected to play an
important role in the global evolution. In this gauge the equations reduce to a
coupled hyperbolic-elliptic system which is formally singular at the axis. Due
to the rather peculiar properties of the system, the local in time existence
has proved to resist analysis by standard methods. To analyze the principal
part of the equations, which may represent the main source of the difficulties,
we study linear perturbation around the flat Minkowski solution in this gauge.
In this article we solve this linearized system explicitly in terms of integral
transformations in a remarkable simple form. This representation is well suited
to obtain useful estimates to apply in the non-linear case.
