  The questions on the presense and quantitative role of the constant terms in
the real part of the high-energy photon-nucleon and photon-nucleus amplitudes
representing the contribution of the non-Regge (the fixed j=0-pole)
singularities in the finite-energy sum rules (FESR) for the photoabsorption
cross sections on nucleons and the lightest atomic nuclei are discussed and new
testable relations are presented for relevant combinations of the Compton
scattering amplitudes.
