Summary. We consider the vectorial approach to the binary Darboux transformations for the Kadomtsev-Petviashvili hierarchy in its Zakharov-Shabat formulation. We obtain explicit formulae for the Darboux transformed potentials in terms of Grammian type determinants. We also study the n-th Gel’fand Dickey hierarchy introducing spectral operators and obtaining similar results. We reduce the above-mentioned results to the Kadomtsev-Petviashvili I and II real forms, obtaining corresponding vectorial Darboux transformations. In particular for the Kadomtsev-Petviashvili I hierarchy, we get the line soliton, the lump solution, and the Johnson-Thompson lump, and the corresponding determinant formulae for the nonlinear superposition of several of them. For Kadomtsev-Petviashvili set describing the motion of strings in the plane. We also consider the I and II real forms for the Gel’fand-Dickey hierarchies obtaining the vectorial Darboux transformation in both cases.

For any two arbitrary positive integers “n” and “m”, using the mth KdV hierarchy and the (n+m)th KdV hierarchy as building blocks, we are able to construct another integrable hicrarchy (referred to as the (n, m)th KdV hierarchy). The W-algebra associated to the second Hamiltonian structure of the (n, m)th KdV hierarchy (called W (n, m) algebra) is isomorphic via a Miura map to the direct sum of a Wm-algebra, a Wn+m-algebra and an additional U (1) current algebra. In turn, from the latter, we can always construct a representation of a W∞-algebra.

A proper bilinear form is proposed for the N=1 supersymmetric modified Korteweg–de Vries equation. The bilinear Backlund transformation for this system is constructed. As applications, some solutions are presented for it.

Sequences of Levy transformations for the Darboux system of conjugates nets in multidimensions are studied. We show that after a suitable number of Levy transformations, with at least a Levy transformation in each direction, we get closed formulae in terms of multi-Wronski determinants. These formulae are for the tangent vectors, Lame coefficients, rotation coefficients and points of the surface.

A multicomponent KdV system, Kupershmidt’s coupled KdV equation, is considered. The system is shown to be a modification of some system, and it can be modified.