A note on Gevrey regular KAM theory and the inverse approximation lemma

A theorem of the type Kolmogorov-Arnold-Moser holds for small real analytic perturbations of a family of parallel flows on a torus, yielding conjugacies to Diophantine parallel flows that depend Whitney-Gevrey regularly on parameters. This follows from an adaptation of the inverse approximation lemma. The method of proof is general and extends to a wide range of problems of KAM type.