BEC with two ground-state filling rates for BCS Cooper-pairs

We report the effects on the thermodynamic properties of a 3D Bose gas caused by a temperature dependent energy gap between the ground state and the first excited state of the energy spectrum of the particles constituting the Bose gas which behaves like an ideal Bose gas when the gap is absent but whose properties are very different when it is present. Explicit formulae are given for the critical temperature, the condensate fraction, the internal energy and the isochoric specific heat, which are calculated for three different gaps which abruptly go to zero at a temperature $T_B$, as well as for the damped counterparts with a smoothed drop to zero. In particular, for the undamped BCS (Bardeen, Cooper and Schrieffer) gap it is observed that the Bose-Einstein condensation (BEC) critical temperature $T_c$ is equal to that of the ideal Bose gas $T_0$, for every $T_B \leq T_0$; surprisingly, the condensate fraction presents two different filling rates of the ground state, one at $T_c = T_0$ and another with a higher rate at $T_B < T_0$, suggesting something like a {\it two-step} BEC; also, its specific heat shows a finite jump at $T_c$ and a divergence at $T_B$ with a critical exponent $\alpha=1/2$, which is inherited from the divergence of the temperature derivative of the BCS gap at $T_B$.