  Perturbation theory in geometric theories of gravitation is a gauge theory of
symmetric tensors defined on a Lorentzian manifold (the background spacetime).
The gauge freedom makes uniqueness problems in perturbation theory particularly
hard as one needs to understand in depth the process of gauge fixing before
attempting any uniqueness proof. This is the first paper of a series of two
aimed at deriving an existence and uniqueness result for rigidly rotating stars
to second order in perturbation theory in General Relativity. A necessary step
is to show the existence of a suitable choice of gauge and to understand the
differentiability and regularity properties of the resulting gauge tensors in
some "canonical form", particularly at the centre of the star. With a wider
range of applications in mind, in this paper we analyse the fixing and
regularity problem in a more general setting. In particular we tackle the
problem of the Hodge-type decomposition into scalar, vector and tensor
components on spheres of symmetric and axially symmetric tensors with finite
differentiability down to the origin, exploiting a strategy in which the loss
of differentiability is as low as possible. Our primary interest, and main
result, is to show that stationary and axially symmetric second order
perturbations around static and spherically symmetric background configurations
can indeed be rendered in the usual "canonical form" used in the literature
while loosing only one degree of differentiability and keeping all relevant
quantities bounded near the origin.
