  Corrections to energy levels in light muonic atoms are investigated in order
$\alpha^2(Z\alpha)^4m$. 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 ($2p_{1/2}-2s_{1/2}$) and
fine-structure intervals ($2p_{3/2}-2p_{1/2}$) in muonic hydrogen, deuterium,
and muonic helium ions.
