The ${\alpha }^2(Z\alpha {)}^{4}m$ contributions to the Lamb shift and the fine structure in light muonic atoms

Corrections to energy levels in light muonic atoms are investigated in order ${\alpha }^2(Z\alpha {)}^{4}m$. We pay attention to corrections which are specific for muonic atoms and include the electron vacuum polarization loop. In particular, we calculate relativistic and relativistic-recoil two-loop electron vacuum polarization contributions. The results are obtained for the levels with $n=1,2$ and in particular for the Lamb shift ($2p_{1\mathrm{/}2}-2s_{1\mathrm{/}2}$) and fine-structure intervals ($2p_{3\mathrm{/}2}-2p_{1\mathrm{/}2}$) in muonic hydrogen, deuterium, and muonic helium ions.