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We study the muonic nuclear-polarization corrections, which provide level shifts due to the two-photon exchange process between a bound muon and a nucleus. We choose $^{16}\mathrm{O}$ as a nucleus for the demonstration of the amount and the property of the nuclear polarization in the muonic atoms. The nuclear states of $^{16}\mathrm{O}$ are constructed in the random-phase approximation including the negative-energy states based on the relativistic mean field model. The spatial components of the transition current have large couplings between positive- and negative-energy states. As a result, the contribution from the negative-energy states of nucleus to the nuclear-polarization correction is found to be significant and also essential to achieve gauge invariance. The nuclear-polarization effect in muonic $^{16}\mathrm{O}$ is also calculated using the collective model. We find that the nonrelativistic nuclear model with the effective mass provides similar results as the relativistic one.

We calculate nuclear-polarization energy shifts in muonic ${}^{208}\mathrm{Pb}.$ We employ a relativistic field-theoretical calculation and evaluate the ladder, cross, and seagull terms of the two-photon exchange diagrams in both the Feynman and Coulomb gauges. Gauge independence is very well satisfied with the calculated nuclear-polarization energies. Using these results, we analyze fine-structure splitting energies of muonic ${}^{208}\mathrm{Pb}$ because of the presence of the persisting discrepancies between experiment and calculation. The present nuclear-polarization energies explain about half of the anomaly in the $\ensuremath{\Delta}2p$ fine-structure splitting energy, and only one-fourth of the anomaly in the $\ensuremath{\Delta}3p$ fine-structure splitting energy.