  Theory of superconductivity in two-band non-adiabatic systems with strong
electron correlations in the linear approximation over non-adiabaticity is
built in the article. Having assumed weak electron-phonon interaction
analytical expressions for the vertex functions and "intersecting " functions
for each energy band are obtained. Mass operators of Green functions with
account of diagrams with intersection of two lines of electron- phonon
interaction (exceeding the frameworks of Migdal theorem) are derived as well as
main equations of theory of superconductivity in two-band systems. Analytical
formulas for the temperature of superconducting transition Tc are obtained as
well as renormalization of the quantities which enter these formulas because of
effects of non-adiabaticity and strong electron correlations. Dependence of Tc
and coefficient of isotope effect on Migdal parameter m is studied. Overlapping
of energy bands on Fermi surface with effects of non-adiabaticity at low values
of transferred momentum is shown to obtain high values of Tc even assuming weak
electron-phonon interaction.
