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Contributions to the energy levels in light muonic atoms and, in particular,to the Lamb shift fall into a few well-distinguished classes. The relateddiagrams are calculated using different approaches. In particular, there is aspecific kind of non-relativistic contributions. Here we consider suchcorrections to the Lamb shift in order $\alpha^5m_\mu$. These contributions aredue to free vacuum polarization loops as well as to various effects oflight-by-light scattering. The closed loop in the related diagrams is anelectronic one, which allows a non-relativistic consideration of the muon. Bothkinds of contributions have been known for a while, however, the resultsobtained up to date are only partial ones. We complete a calculation of the$\alpha^5m_\mu$ contributions for muonic hydrogen. The results are alsoadjusted for muonic deuterium and muonic helium ion.

Non-relativistic contributions in order $\alpha^5m_\mu c^2$ to the Lamb shift in muonic hydrogen, deuterium and helium ion