The reversible context 2 in KAM theory: the first steps

The reversible context 2 in KAM theory refers to the situation where $\dim\mathop{\rm Fix} G<\frac{1}{2}\mathop{\rm codim}{\mathcal T}$, here $\mathop{\rm Fix} G$ is the fixed point manifold of the reversing involution $G$ and $\mathcal T$ is the invariant torus one deals with. Up to now, this context has been entirely unexplored. We obtain a first result on the persistence of invariant tori in the reversible context 2 (for the particular case where $\dim\mathop{\rm Fix} G=0$) using J. Moser's modifying terms theorem of 1967.