Whitney Smooth Families of Invariant Tori within the Reversible Context 2 of KAM Theory

We prove a general theorem on the persistence of Whitney $C^\infty$-smooth families of invariant tori in the reversible context 2 of KAM theory. This context refers to the situation where $\dim \text{Fix}\,G < (\text{codim}\,\mathcal{T})/2$, where $\text{Fix}\,G$ is the fixed point manifold of the reversing involution $G$ and $\mathcal{T}$ is the invariant torus in question. Our result is obtained as a corollary of the theorem by H. W. Broer, M.-C. Ciocci, H. Hansmann, and A. Vanderbauwhede (2009) concerning quasi-periodic stability of invariant tori with singular “normal” matrices in reversible systems.