(see [@STER] and references therein).

Given that the differential operators appearing in the perturbative treatment of a problem are those defined on the background, the theory is considerably simplified when the latter is spherical. Generic time dependent perturbations of the rotating star (parametrized by a dummy parameter $\lambda$ and describing oscillations) are then built on top of the stationary axisymmetric perturbations in $\Omega$. Clearly, in this approach any interesting physics requires non–linear perturbations, as at least terms of order $\lambda\Omega$ need to be considered. A similar approach could be used to study perturbations of the slowly rotating collapse, even if in specific cases [@CPM; @seidel; @gundlach] the perturbative expansion depends by one parameter only.

Classical studies in the literature have not analysed in full the gauge dependence and gauge invariance of the non-linear perturbation theory. For example, in [@CPM] the second order perturbations are treated in a gauge invariant fashion on top of the first order perturbation in a given specific gauge. The perturbation variables used are therefore non gauge invariant under a complete second order gauge transformation [@BMMS; @FW], but only invariant under “first order transformations acting at second order” [@CPM]. While this may be perfectly satisfactory from the 