  The finite-nuclear size correction to the fine structure of muonic atoms are
considered. The procedure for the analytical calculation of the energies and
wave functions has been derived in a homogeneously charged sphere nuclear
charge distribution approximation. The finite-nuclear size effect was
calculated in a first few orders of the perturbation energy, with the accurate
estimations of the convergence. Finally, we present energies of the low-lying
electronic and muonic states with the finite-nuclear size correction,
calculated analytically, for ${}^{185}_{75}\text{Re}$ ion.
