  The solution generating methods discovered earlier for integrable reductions
of Einstein's and Einstein - Maxwell field equations (such as soliton
generating techniques, B$\ddot{a}$cklund or symmetry transformations and other
group-theoretical methods) can be described explicitly as transformations of
especially defined "coordinates" in the infinite-dimensional solution spaces of
these equations. In general, the role of such "coordi\-nates", which
characterize every local solution, can be performed by the monodromy data of
the fundamental solutions of the corresponding spectral problems. However for
large subclasses of fields, these can be the values of the Ernst potentials on
the boundaries which consist of such degenerate orbits of the space-time
isometry group, in which neighbourhood the space-time geometry and
electromagnetic fields possess a regular behaviour. In this paper,
transformations of such "coordinates", corresponding to different known
solution generating procedures are described by simple enough algebraic
expressions which do not need any particular choice of the initial (background)
solution. Explicit forms of these transformations allow us to find the
interrelations between the sets of free parameters, which arise in different
solution generating procedures, as well as to determine some physical and
geometrical properties of each generating solution even before a detail
calculations of all its components.
