In series of works resonance excitation of the $^8$$^3$Kr first nuclear level (E = 9.4 keV) by solar axions formed via the Primakoff mechanism is sought. The $\gamma$- and X-ray photons, the conversion and Auger electrons arising from the excited-level relaxation are detected with a gas proportional counter of a low-background detector in the underground Baksan Neutrino Observatory. The following experimental constraint is obtained for the product of the axion–photon coupling constant and the axion mass: |g$_A$$_\gamma$×m$_A$| ≤ 6.3×10$^-$$^7$ and model independent upper limit on the combination of isoscalar and isovector axion-nucleon couplings |g$_3$−g$_0$|≤8.4×10$^-$$^7$. In the framework of the hadronic-axion model, this corresponds to a new axion-mass constraint of m$_A$ ≤ 12.7 eV at 95% C.L.