Algebraic construction of the Darboux matrix revisited

We present algebraic construction of Darboux matrices for 1+1dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case. We discuss different approaches to the Darboux-Bäcklund transformation, based on different λ-dependencies of the Darboux matrix: polynomial, sum of partial fractions, or the transfer matrix form. We derive symmetric N -soliton formulas in the general case. The matrix spectral parameter and dressing actions in loop groups are also discussed. We describe reductions to twisted loop groups, unitary reductions, the matrix Lax pair for the KdV equation and reductions of chiral models (harmonic maps) to SU(n) and to Grassmann spaces. We show that in the KdV case the nilpotent Darboux matrix generates the binary Darboux transformation. The paper is intended as a review of known results (usually presented in a novel context) but some new results are included as well, e.g., general compact formulas for N -soliton surfaces and linear and bilinear constraints on the nonisospectral Lax pair matrices which are preserved by Darboux transformations. PACS Numbers: 02.30.Ik, 03.50.Kk, 05.45Yv. MSC 2000: 37K35, 37K30, 37K25, 35Q53, 22E67.