Microemulsions are explored using the self-consistent field approach. We consider a balanced model that features two solvents of similar size and a symmetric surfactant. Interaction parameter χ and surfactant concentration φsb complement the model definition. The phase diagram in χ-φsb coordinates is known to feature two lines of critical points, the Scott and Leibler lines. Only upon imposing a finite distance between the interfaces, we observe that the Scott line meets the Leibler line. We refer to this as a Lifshitz point (LP) for real systems. We add regions that are relevant for microemulsions to this phase diagram by considering the saturation line, which connects (χ, φsb)-points for which the interface becomes tensionless. Crossing this line implies a first-order phase transition as internal interfaces develop, characteristic for one-phase microemulsions. The saturation line ends at the so-called microemulsion point (MP). The MP is shown to connect with the LP by a line of MP-like critical points, found by searching for a "MP" while the distance between interfaces is fixed. A pair of binodal lines that envelop the three-phase (Winsor III) microemulsion region is shown to connect to the MP. The cohesiveness of the middle phase in Winsor III is related to non-monotonic, inverse DLVO-type interaction curves between the surfactant-loaded tensionless interfaces. The mean and Gaussian bending modulus, relevant for the shape fluctuations and the topology of interfaces, respectively, are evaluated along the saturation line. Near the MP, both rigidities are positive and vanish in a power-law fashion with coefficient unity at the MP. Overseeing these results proves that the MP has a pivoting role in the physics of microemulsions.