  An effective description of the inverse spectral data corresponding to the
real periodic and quasiperiodic solutions for the sine-gordon equation is
obtained. In particular, the explicit formula for the so-called topological
charge of the solutions is found and proved. As it was understood already 20
years ago, it is very hard to extract any formula for this quantity from the
Theta-functional expressions. A new method was developed by the authors for
this goal. In the appendix 3 an analog of the Fourier-Laurent integral
transform on the Riemann surfaces is defined, based on the constructions of the
Soliton Theory. It is a natural continuous analog of the discrete
Krichever-Novikov bases developed in the late 80s for the needs of the Operator
Quantization of String Theory.
