  This paper is part of a program that aims to understand the connection
between the emergence of chaotic behaviour in dynamical systems in relation
with the multi-valuedness of the solutions as functions of complex time $\tau$.
In this work we consider a family of systems whose solutions can be expressed
as the inversion of a single hyperelliptic integral. The associated Riemann
surface ${\mathcal R}$ is known to be an infinitely sheeted covering of the
complex time plane, ramified at an infinite set of points whose projection in
the $\tau$-plane is dense. The main novelty of this paper is that the
geometrical structure of these infinitely sheeted Riemann surfaces is described
in great detail, which allows to study global properties of the flow such as
asymptotic behaviour of the solutions, periodic orbits and their stability or
sensitive dependence on initial conditions. The results are then compared with
a numerical integration of the equations of motion. Following the recent
approach of Calogero, the real time trajectories of the system are given by
paths on $\mathcal R$ that are projected to a circle on the complex plane
$\tau$. Due to the branching of $\mathcal R$, the solutions may have different
periods or may not be periodic at all.
