Using neutron scattering, we investigate the structure at local scale of very dilute solutions of a quasiternary surfactant system. For such systems consisting of surfactant and cosurfactant in brine, a critical line is often found in the very dilute range of the phase diagram where the L3 sponge domain is continuously connected to the micellar domain. Our purpose is to examine the evolution of the morphology of the surfactant aggregates close to this critical line. Starting in the L3 sponge domain and approaching the transition line, we find that the neutron scattering patterns deviate more and more from that of an infinite bilayer. Furthermore, far away on the other side of the critical line, the neutron scattering patterns reveal that the initial bilayers have been replaced by long 1D cylindrical micelles. These findings are discussed in terms of the two different mechanisms that have been considered in the literature as being possibly at the origin of the critical behaviour: the symmetry breaking of the sponge and the spontaneous tearing of the bilayer