  By studying periodic points for rational maps on $\bm{C}^d$ with $p$
invariants, we show that they form an invariant variety of dimension $p$ if the
periodicity conditions are `fully correlated', and a set of isolated points if
the conditions are `uncorrelated'. We present many examples of the invariant
varieties in the case of integrable maps. Moreover we prove that an invariant
variety and a set of isolated points do not exist in one map simultaneously.
