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.