  We propose a version of the classical Artin approximation which allows to
perturb the variables of the approximated solution. Namely, it is possible to
approximate a formal solution of a Nash equation by a Nash solution in a
compatible way with a given Nash change of variables. This results is closely
related to the so-called nested Artin approximation and becomes false in the
analytic setting. We provide local and global version of this approximation in
real and complex geometry together with an application to the Right-Left
equivalence of Nash maps.
