The experimental data analysis we have carried out shows that there are at the least two additional experimental dataOf which the hypothesis Of the A-isobar localized state (A-ball) formation can be used. Possible mechanism of A-isobar localized state formation in nuclear reaction is considered. A set of main A-isobar localized state characteristics is obtained and the model of II, y-production in pnucleus and nucleus-for the nucleus collisions through A-localized state formation is suggested. The First, it is the experimental observation of the abnormal results of the calculations show that the theory successfully explains the characteristics of narrow resonance (EP = 350MeV, r 10 MeV), observed recently in proton-nucleus collisions as well as the effect of stability of the pion spectra slope parameter near the absolute threshold. The experiments for checking the outlined theoretical approach are suggested.