From experiments with ice or metal crystals, in the vicinity of their crystal/liquid/vapor triple points, it is known that melting of crystals starts on their surfaces and is anisotropic. It is shown here by direct observations under an optical microscope that this anisotropic surface melting phenomenon occurs also in lyotropic systems. In the case of C12EO2/water mixture, it takes place in the vicinity of the peritectic Pn3m/L3/L1 triple point. Above the peritectic triple point, where the Pn3m and L1 phases coexist in the bulk, the surface of a Pn3m-in-L1 crystal is composed of (111)-type facets surrounded by rough surfaces. The angular junction suggests that rough surfaces are wet by a L3-like layer while facets stay "dry". This is analogous to the pre-melting at rough surfaces in solid crystals. Upon cooling below the peritectic triple point, where L3 and L1 phases coexist in the bulk, a thick layer of the L3 phase grows from the pre-melted, rough Pn3m/L1 interface. Simultaneously, facets stay dry and their radius decreases. In this tri-phasic configuration, stable in a narrow temperature range, the L3/L1 and L3/Pn3m interfaces have shapes of constant mean curvature surfaces having common borders: edges of facets.