  We consider an Abel polynomial differential equation. For two given points a
and b, the "Poincare mapping" of the equation transforms the values of its
solution at a into their values at b. In this article, we study global analytic
properties of the Poincare mapping, in particular, its analytic continuation,
its singularities and its fixed points. On one side, we give a general
description of singularities of the Poincare mapping, and of its analytic
continuation. On the other side, we study in detail the structure of the
Poincare mapping for a local model near a simple fixed singularity, where an
explicit solution can be written. Yet, the global analytic structure (in
particular the ramification) of the solutions and of the Poincare mapping in
this case is fairly complicated, and, in our view highly instructive. For a
given degree of the coefficients we produce examples with an infinite number of
complex periodic solutions and analyze their mutual position and branching. Let
us remind that Pugh's problem, which is closely related to the classical
Hilbert's 16th problem, asks for the existence of a bound to the number of real
isolated periodic solutions.
