  In this note we will discuss a potentially interesting extension of some
recent results on primitive solutions to completely integrable partial
differential equations. We will discuss a family distributions that are
holomorphic on the Riemann sphere except on the singular sets homeomorphic to a
Cantor set or Sierpinski gasket. These distributions allow us to produce
solutions to the Kadomtsev--Petviashvili equation. These distributions are
limits of families of rational functions that can also be associated with
holomorphic line bundles on surfaces with a finite number of doubly degenerate
singular points. We conjecture that a subset of these distributions can be used
to formulate a definition of a holomorphic line bundle on some surfaces that
are homeomorphic to spheres except where they become doubly degenerate on
singular sets homeomorphic to a Cantor set or Sierpinski gasket.
