Bianchi–Li–Backlund transformation in spaces of constant curvature $H^3(-1)$ and $S^3(1)$

Bianchi–Lie–Backlund transformation in space forms $H^3(-1)$ (Poincare model of Lobachevsky space at the upper half-plane) and $S^3(1)$ (spherical space with the Riemann metric) are considered. The conditions defining the transformation in global coordinates and the corresponding differential equations of surfaces of constant external curvature are derived.