  A diagonal metric sum_{i=1}^n g_{ii} dx_i^2 is termed Guichard_k if
sum_{i=1}^{n-k}g_{ii}-sum_{i=n-k+1}^n g_{ii}=0. A hypersurface in R^{n+1} is
isothermic_k if it admits line of curvature co-ordinates such that its induced
metric is Guichard_k. Isothermic_1 surfaces in R^3 are the classical isothermic
surfaces in R^3. Both isothermic_k hypersurfaces in R^{n+1} and Guichard_k
orthogonal co-ordinate systems on R^n are invariant under conformal
transformations. A sequence of n isothermic_k hypersurfaces in R^{n+1}
(Guichard_k orthogonal co-ordinate systems on R^n resp.) is called a Combescure
sequence if the consecutive hypersurfaces (orthogonal co-ordinate systems
resp.) are related by Combescure transformations. We give a correspondence
between Combescure sequences of Guichard_k orthogonal co-ordinate systems on
R^n and solutions of the O(2n-k,k)/O(n)xO(n-k,k)-system, and a correspondence
between Combescure sequences of isothermic_k hypersurfaces in R^{n+1} and
solutions of the O(2n+1-k,k)/O(n+1)xO(n-k,k)-system, both being integrable
systems. Methods from soliton theory can therefore be used to construct
Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces
of these geometric objects and their loop group symmetries.
