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We analyze a class of the BCS model, whose free dispersion relation isnon-vanishing, under the influence of imaginary magnetic field at positivetemperature. The magnitude of the negative coupling constant must be small butis allowed to be independent of the temperature and the imaginary magneticfield. The infinite-volume limit of the free energy density is characterized. Aspontaneous symmetry breaking and an off-diagonal long range order are provedto occur only in high temperatures. This is because the gap equation in thismodel has a positive solution only if the temperature is higher than a criticalvalue. The proof is based on a double-scale integration of the Grassmannintegral formulation. In this scheme we integrate with the infrared covariancefirst and with the ultra-violet covariance afterwards. As the other focus, westudy geometric properties of the phase boundaries, which are periodic copiesof a closed curve in the two-dimensional space of the temperature and the realtime variable. Here we adopt the real time variable in place of the temperaturetimes the imaginary magnetic field by considering its relevance withincontemporary physics of dynamical phase transition at positive temperature. Asthe main result, we show that for any choice of a non-vanishing free dispersionrelation the representative curve of the phase boundaries has only one localminimum point, or in other words the phase boundaries do not oscillate withtemperature, if and only if the minimum of the magnitude of the free dispersionrelation over the maximum is larger than the critical value$\sqrt{17-12\sqrt{2}}$.

Superconducting phase in the BCS model with imaginary magnetic field. III. Non-vanishing free dispersion relations