  For axially symmetric solutions of Einstein equations there exists a gauge
which has the remarkable property that the total mass can be written as a
conserved, positive definite, integral on the spacelike slices. The mass
integral provides a nonlinear control of the variables along the whole
evolution. In this gauge, Einstein equations reduce to a coupled
hyperbolic-elliptic system which is formally singular at the axis. As a first
step in analyzing this system of equations we study linear perturbations on
flat background. We prove that the linear equations reduce to a very simple
system of equations which provide, thought the mass formula, useful insight
into the structure of the full system. However, the singular behavior of the
coefficients at the axis makes the study of this linear system difficult from
the analytical point of view. In order to understand the behavior of the
solutions, we study the numerical evolution of them. We provide strong
numerical evidence that the system is well-posed and that its solutions have
the expected behavior. Finally, this linear system allows us to formulate a
model problem which is physically interesting by itself, since it is connected
with the linear stability of black holes solutions in axial symmetry. This
model can contribute significantly to solve the nonlinear problem and at the
same time it appears to be tractable.
