Derivation of invariant varieties of periodic points from singularity confinement in the case of Toda map

•Invariant varieties of periodic points (IVPP) for the 3-point Toda map are derived.•An algorithm is developed to derive IVPPs for maps with less number of invariants.•If IVPPs exist, the singularity confinement is shown to be an integrability test.

We have shown in [1] that the invariant varieties of periodic points (IVPP) of all periods of some higher dimensional rational maps can be derived, iteratively, from the singularity confinement (SC). We generalize this algorithm, in this paper, to apply to any birational map, which has more invariants than the half of the dimension.