  Topology of the Generic Hamiltonian Dynamical Systems on the Riemann Surfaces
given by the real part of the generic holomorphic 1-forms, is studied. Our
approach is based on the notion of Transversal Canonical Basis of Cycles (TCB).
This approach allows us to present a convenient combinatorial model of the
whole topology of the flow, especially effective for g=2. A maximal abelian
covering over the Riemann Surface is needed here. The complete combinatorial
model of the flow is constructed. It consists of the Plane Diagram and g
straight line flows in the 2-tori ''with obstacles''. The Fundamental Semigroup
of positive closed paths transversal to foliation, and the topological
characteristics of trajectories, are studied. This work contains an improved
exposition of the results presented in the authors recent preprint (arXiv
math.GT/0105338) and new results calculating all TCB in the 2-torus with
obstacle, in terms of Continued Fractions.
