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