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Spectral problems of the third order (matrix and scalar) and associated systems of non-linear equations are studied. The technique to construct the solutions is developed for possible reduction constraints. The Dodd-Bullough-Mikhailov.Hirota-Satsuma, Boussinesq. Sawada-Kotera. Kaup-Kupershmidt and less popular reductions of nonlinear systems are treated by means of the Darboux transformation method with the covariant bilinear conditions for the given reduction. Properties of some new soliton-like solutions are discussed. The generalization of Darboux lransformation theorems for the reductions corresponding to the above mentioned equations in two spatial and one temporal dimensions are given.

Lax pairs with operator valued coefficients, which are explicitly connected by means of an additional condition, are considered. This condition is proved to be covariant with respect to the Darboux transformation of a general form. Nonlinear equations arising from the compatibility condition of the Lax pairs in the matrix case include, in particular, Nahm equations, Volterra, Bogoyavlenskii and Toda lattices. The examples of another one-, two-and multi-field lattice equations are also presented.PACS 02.30.Ik, 05.45.Yv