Sum Rules for Total Interaction Cross Sections of Relativistic Elementary Atoms with Atoms of Matter Up to Terms of Order $\alpha^2$

It is shown that $\alpha^2$-term of sum rules for the total cross sections of interaction of elementary atoms with matter ones, obtained in ref.[Denisenko K. and Mrowczynski S., Phys. Rev. D36 (1987) 1529] is wrong. New sum rules valid up to terms of order $\alpha^2$ are derived.