The phase diagram of the ternary surfactant system tetradecyldimethylamine oxide (TDMAO)/HCl/1-hexanol/water shows with increasing cosurfactant concentration an L-1 phase, two L-alpha phases (a vesicle phase L-alpha 1 and a stacked bilayer phase L-alpha h), and an L-3 phase, which are separated by the corresponding two-phase regions L-1/L-alpha and L-alpha/L-3. In this investigation, the system was studied where some of the TDMAO was substituted by the protonated TDMAO. Under these conditions, one finds for constant surfactant concentration of 100 mM TDMAO a micellar L1 phase, an L-alpha 1 phase (consisting of multilamellar vesicles), and an interesting isotropic L-1* phase in the middle of the L-1/L-alpha two-phase region. The L-1* phase exists at intermediate degrees of charging of 30-60% and for 40-120 mM TDMAO and 70-140 mM hexanol concentration. At surfactant concentrations less than 80 mM the L-1*-phase borders directly on the L-1 phase. The phase transition between the L-1 phase and the L-1* phase was detected by electric conductivity and rheological measurements. The conductivity values show a sharp drop at the L-1/L-1* transition, and the zero shear viscosity of the L-1* phase is much lower than in L-1 phase. The form and size of the aggregates in L-1* were detected with FF-TEM and SANS. This phase contains small unilamellar vesicles (SUV) of about 10 nm and some large multilamellar vesicles with diameters up to 500 nm. The system exhibits another peculiarity. For 100 mM surfactant, the clear L-alpha 1-phase exists only at chargings below 30%. With oscillating rheological measurements a parallel development of the storage modulus G' and the loss modulus G " was observed. Both moduli are frequency independent and the system possesses a yield stress. The storage modulus is a magnitude larger than the loss modulus.