  The set of real finite-gap Sine-Gordon solutions corresponding to a fixed
spectral curve consists of several connected components. A simple explicit
description of these components obtained by the authors recently is used to
study the consequences of this property. In particular this description allows
to calculate the topological charge of solutions (the averaging of the
$x$-derivative of the potential) and to show that the averaging of other
standard conservation laws is the same for all components.
