  We investigate the structure of fully non-linear P.D.E.'s in holomorphic
functions, with emphasis on the functorial generalisation of so called
"irregular" O.D.E.'s. Highlights are an implicit function theorem removing the
perturbation conditions of Nash-Moser type, best possible existence results
when the singularity of the linearised P.D.E. is at worst bi-dimensional, and
various, again optimal, corollaries on existence of centre manifolds and
conjugation to normal form of 3-dimensional vector fields.
