  An implicit fundamental assumption in relativistic perturbation theory is
that there exists a parametric family of spacetimes that can be Taylor expanded
around a background. The choice of the latter is crucial to obtain a manageable
theory, so that it is sometime convenient to construct a perturbative formalism
based on two (or more) parameters. The study of perturbations of rotating stars
is a good example: in this case one can treat the stationary axisymmetric star
using a slow rotation approximation (expansion in the angular velocity Omega),
so that the background is spherical. Generic perturbations of the rotating star
(say parametrized by lambda) are then built on top of the axisymmetric
perturbations in Omega. Clearly, any interesting physics requires non-linear
perturbations, as at least terms lambda Omega need to be considered. In this
paper we analyse the gauge dependence of non-linear perturbations depending on
two parameters, derive explicit higher order gauge transformation rules, and
define gauge invariance. The formalism is completely general and can be used in
different applications of general relativity or any other spacetime theory.
