  Topology of Foliations of the Riemann Surfaces given by the real part of
generic holomorphic 1-forms, is studied. Our approach is based on the notion of
Transversal Canonical Basis of Cycles (TCB) instead of using just one closed
transversal curve as in the classical approach of the ergodic theory. In some
cases the TCB 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 provided by the Abel Map, plays a key role in
this work. The behavior of our system in the Fundamental Domain of that
covering can be easily described in the sphere with $g$ holes. It leads to the
Plane Diagram of our system. The complete combinatorial model of the flow is
constructed. It is based on the Plane Diagram and g straight line flows in the
planes corresponding to the $g$ canonically adjoint pairs of cycles in the
Transversal Canonical Basis. These pairs do not cross each other. Making cuts
along them, we come to the maximal abelian fundamental domain (associated with
Abel Map and Theta-functions) instead of the standard $4g$-gon in the
Hyperbolic Plane and its beautiful "flat" analogs which people used for the
study of geodesics of the flat metrics with singularities. Topological
splitting of the flow into torical pieces is constructed. Several mistakes are
corrected. Algebraic description of transversal canonical bases on the torus
with obstacles is given. The appendix is extended: a reference to a work of G.
Levitt is added which allowed to prove that every foliation of this class
admits a TCB.
