KAM theory for lower dimensional tori within the reversible context 2

The reversible context 2 in KAM theory refers to the situation where $\mathrm{dim}\,\mathrm{Fix}\, G<\frac{1}{2}\mathrm{codim}\,\mathcal{T}$, here $\mathrm{Fix}\, G$ is the fixed point manifold of the reversing involution $G$ and $\mathcal{T}$ is the invariant torus one deals with. Up to now, the persistence of invariant tori in the reversible context 2 has been only explored in the extreme particular case where $\mathrm{dim}\,\mathrm{Fix}\,G=0$ [M. B. Sevryuk, Regul. Chaotic Dyn. 16 (2011), no. 1–2, 24–38]. We obtain a KAM-type result for the reversible context 2 in the general situation where the dimension of $\mathrm{Fix}\, G$ is arbitrary. As in the case where $\mathrm{dim}\,\mathrm{Fix}\, G=0$, the main technical tool is J. Moser's modifying terms theorem of 1967.