An investigation is made into the capillary-gravity waves which arise on the interface of two fluids and which are formed by the interaction of the first two harmonics of the motion. The problem is transformed into a nonlinear operator equation between suitable function spaces which is shown to be invariant under certain group actions. The infinite dimensional problem is reduced, by the classical procedure of Lyapunov–Schmidt, to a finite system of polynomial equations, known as the bifurcation equations. Because these equations inherit the symmetry properties of the original operator, it is possible to make quite specific statements concerning their structure, thus rendering their analysis easier. Solutions to the equations are sought, both in the cases of exact and of near-resonance. A wide variety of solutions is found depending on the values of the parameters: both simple, multiple and secondary bifurcations may occur, and in addition, there may exist isolated solution curves.