For a family of reversible vector fields having a fixed point at the origin, we present the problem where , at criticality, the derivative at the origin has a multiple 0 eigenvalue with a 4 x 4 Jordan block. This is a codimension 2 singularity for reversible vector fields. This case happens in the water-wave problem for Bond number 1/3 and Froude number 1.We study the persistence of all known phenomena on the codimension one curves (in the parameter plane), especially concerning homoclinic orbits. One of these unfoldings is the 1:1 resonance Hopf bifurcation. The study strongly relies upon the knowledge of the reversible normal forms associated with the 4 X 4 Jordan block, and the unfolded situations , together with appropriate scalings.