The influence of the overlapping of energy bands on the superconductivity of the localized Bose-condensated pairs at $T=0$

A system of equations for the order parameters $\Delta _{nm}$ and the chemical potential $\mu $ of the two-band superconductors with a low carrier density at $T=0$ is obtained. The limit case $\mu _n<0$ corresponding to the Bose-condensation of the localized pairs in the both bands is considered. The all possible interband and intraband electron-electron interactions are taking into account. Provided $\Delta _{nm}^2/\mu _{n,m}\ll 1$ the system of equations mentioned above is lead to the system of the linear equations. The analitical solutions of these equations are obtained. The dependence of the quantities $\Delta _{nm}$ and $\mu$ on the electron-electron interaction and the carrier density is analysed.