Singular matrix Darboux transformations in the inverse scattering method
  Singular Darboux transformations, in contrast to the conventional ones, have a singular matrix as a coefficient before the derivative. We incorporated such transformations into a chain of conventional transformations and presented determinant formulas for the resulting action of the chain. A determinant representation of the Kohlhoff-von Geramb solution to the Marchenko equation is given. 
A. A. Pecheritsin, A. M. Pupasov and Boris F. Samsonov
quant-ph hep-th math-ph math.MP