  We introduce primitive forms with or without higher residue structure and
explore their connection with the flat structures with or without a metric and
integrable hierarchies of KdV type. Just as the classical case of primitive
forms with metric arXiv:1311.1659, the primitive forms without metrics are
constructed as the positive part of the Birkhoff decomposition of formal
oscillatory integrals with respect to the descendent variable. The oscilating
integrals of a primitive form without metric give rise to a hierarchy of
commuting PDE of the KdV type as in the case of primitive forms with metric.
This shall be studied in (II).
