KAM theory for lower dimensional tori within the reversible context 2

The reversible context 2 in KAM theory refers to the situation where dim Fix G < (1/2) codim T, here Fix G is the fixed point manifold of the reversing involution G and 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 dim 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 Fix G is arbitrary. As in the case where dim Fix G = 0, the main technical tool is J. Moser's modifying terms theorem of 1967.