  The nonlinear equations describing all the nonsingular pencils of metrics of
constant Riemannian curvature are derived and the integrability of these
nonlinear equations by the method of inverse scattering problem is proved. It
is proved that all the nonsingular pairs of compatible metrics of constant
Riemannian curvature are described by special integrable reductions of
nonlinear equations defining orthogonal curvilinear coordinate systems in the
spaces of constant curvature.
