On orthogonal curvilinear coordinate systems in constant curvature spaces

We describe a method for constructing an $n$-orthogonal coordinate system in constant curvature spaces. The construction proposed is actually a modification of the Krichever method for producing an orthogonal coordinate system in the $n$-dimensional Euclidean space. To demonstrate how this method works, we construct some examples of orthogonal coordinate systems on the twodimensional sphere and the hyperbolic plane, in the case when the spectral curve is reducible and all irreducible components are isomorphic to a complex projective line.