本文从约化的角度考虑BKP方程族的Pfaffian形式的解，证明了通过施加适当的微分约束，KP方程族的格拉瞬行列式的解很自然的约化为BKP方程族的解。

We consider an energy-dependent version of the third-order scalar Lax operator, thus extending our previous results. Unlike the second-order case it is no longer possible to expand the potentials as arbitrary polynomials in b. We prove that there are exactly four cases. We present Hamiltonian operators, Hamiltonian Miura maps and modifications for two new examples.

We consider the Darboux type transformations for the spectral problems of supersymmetric KdV systems. The supersymmetric analogies of Darboux and Darboux-Levi transformations are established for the spectral problems of Manin-Radul-Mathieu sKdV and Manin-Radul sKdV. Sevetral Backlund transformations are derived for the MRM sKdV and MR sKdV systems.

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.

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.