  We study deformations of free boundary constant mean curvature (CMC)
hypersurfaces whose Jacobi operator is degenerate due to symmetries of the
ambient space. The value of the mean curvature and the ambient metric are
allowed to vary simultaneously, provided that the infinitesimal ambient
symmetries change smoothly. We discuss applications to free boundary CMC disks
and Delaunay annuli in the unit ball of a space form.
