  In the present work, the spin-coefficient formalism of Newman and Penrose is
used to formulate geometric constraints for the existence of Kerr-Schild
groups, i.e. continuous groups of generalized Kerr-Schild transformations. In
addition, by characterizing the geometric structure of the deformed Einstein
tensor of the generalized Kerr-Schild class, restrictions are imposed on the
existence of apparent conservation laws in generic spacetimes, which are
defined via considering special Kerr-Schild currents whose associated
Kerr-Schild vector fields coincide with timelike Killing vector fields of pairs
of stationary background geometries. The feasibility of the derived conditions
is demonstrated by considering concrete, suitably simple models of generalized
Kerr-Schild spacetimes.
