Darboux transformation for the simplest phenomenological brane model

The method of isospectral symmetries is used to integrate the Einstein equations in the presence of a family of parallel 3-branes immersed in an ambient five-dimensional space filled with gravity and a scalar field. In the presence of orbifold symmetry, for a given stationary brane, part of the resulting equations can be reduced to the Schrodinger equation, the discrete non-negative spectrum of which determines a set of particles on the brane. The Schrodinger equation admits a Darboux transformation, which, when using a regular support function, changes the spectrum to a finite number of levels without changing the others. Thus, the described technique allows one to construct finite sets of particles with a given mass distribution. In the approximation of fine-tuning between the brane tension and the value of the vacuum energy density, exact solutions are constructed for a brane with zero curvature. An exact derivation of the generalized Crum formulas is given.