In this work we develop a general procedure for constructing the recursion operators for nonlinear integrable equations admitting Lax representation. Several new examples are given. In particular, we find the recursion operators for some KdVtype systems of integrable equations.

A new integrable sixth-order nonlinear wave equation is discovered by means of the Painlevé analysis, which is equivalent to the Kortewegde Vries equation with a source. A Lax representation and a Bäcklund self-transformation are found of the new equation, and its travelling wave solutions and generalized symmetries are studied.

Gödel-type metrics are introduced and used in producing charged dust solutions in various dimensions. The key ingredient is a (D − 1)-dimensional Riemannian geometry which is then employed in constructing solutions to the Einstein-Maxwell field equations with a dust distribution in D dimensions. The only essential field equation in the procedure turns out to be the source-free Maxwell's equation in the relevant background. Similarly the geodesics of this type of metric are described by the Lorentz force equation for a charged particle in the lower dimensional geometry. It is explicitly shown with several examples that Gödel-type metrics can be used in obtaining exact solutions to various supergravity theories and in constructing spacetimes that contain both closed timelike and closed null curves and that contain neither of these. Among the solutions that can be established using non-flat backgrounds, such as the Tangherlini metrics in (D − 1)-dimensions, there exists a class which can be interpreted as describing black-hole-type objects in a Gödel-like universe.

We formulate stationary axially symmetric (SAS) Einstein–Maxwell fields in the framework of harmonic mappings of Riemannian manifolds and show that the configuration space of the fields is a symmetric space. This result enables us to embed the configuration space into an eight-dimensional flat manifold and formulate SAS Einstein–Maxwell fields as a σ-model. We then give, in a coordinate free way, a Belinskii–Zakharov type of an inverse scattering transform technique for the field equations supplemented by a reduction scheme similar to that of Zakharov–Mikhailov and Mikhailov–Yarimchuk.

We give the conditions for a system of N-coupled Korteweg de Vries ͑KdV͒ type of equations to be integrable. We find the recursion operators of each subclass and give all examples for Nϭ2.