  For an analytic differential system in $\mathbb R^n$ with a periodic orbit,
we will prove that if the system is analytically integrable around the periodic
orbit, i.e. it has $n-1$ functionally independent analytic first integrals
defined in a neighborhood of the periodic orbit, then the system is
analytically equivalent to its Poincar\'e--Dulac type normal form. This result
is an extension for analytic integrable differential systems around a
singularity to the ones around a periodic orbit.
