Minimal energy conformations of amphiphilic monolayers within a microemulsion are investigated. An exploitation of the Euler-Lagrange equations yields the internal forces and torques of the fluid film, which cannot be characterized by a simple surface tension. A numerical scheme reveals various properties of the equilibrium solutions of three families of bicontinuous structures. The influence of the constitutive behaviour is demonstrated not only on the basis of a quadratic bending energy density but also with a non-linear extension which exhibits symmetry breaking. The latter is essential to allow the three-phase coexistence of a microemulsion with excess phases of oil and water and hence the existence of an X point in the phase map.