Interlaced branching equations in the theory of non-linear equations

Necessary conditions for inheriting the interlacing property of a non-linear equation by the branching system are obtained. The case when the pair of linear operators interlacing the equation consists of projections or parametric families of linear operators is considered. New conditions are presented which allow one to reduce the number of the equations in the branching system and extend the range of applications of the method of successive approximations in the branching theory of non-linear equations. Solutions depending on free parameters belonging to certain hypersurfaces in Euclidean spaces are considered. The results obtained add to and extend earlier results on the applications of group analysis in branching theory