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After a short exposition of the theory of local and nonlocal symmetries and conservation laws for systems of PDE’s, results on these and the recursion operator are listed for the system of PDE’s $u_x = vw_x, v_y = uw_y, uv + w_{xx} + w_{yy} = 0$. In between the methods of computation are explained.

The message we tried to bring over may be repeated here again. •Very often a special field of applications needs its own programming, though useful packages exist in all of the ‘Algebra Systems’. •For efficient programming in ‘Symbolic Algebra’ good knowledge of the used system is necessary. For us it is of greatest importance, that the source code of REDUCE is generally available contrary to systems where one buys only the ‘Executable System’. •The programming of functions to your own needs should nevertheless be constructed by using techniques of ‘Software Ingeneering’. Our experience with a ‘WEB’-system, literate programming, fits to some degree into this philosophy.

We present a complete proof that solutions of the WDVV equations in Seiberg-Witten theory may be constructed from root systems. A generalization to weight systems is proposed.

It is proved that the Virasoro algebra can be constructed starting with only two generators L3, L−2 subjected to 5 relations without neither introducing a central element c (charge) nor assuming any symmetry relation. Our proof is supported by substantial symbolic computations in REDUCE.

In this paper we introduce and implement a concept for dealing with mathematical bases of linear spaces and mappings (multi)linear with respect to such bases, in REDUCE (cf. [1]). Using this concept we give some examples how to implement some well known (multi)linear mappings in mathematics with very little effort. Moreover we implement a procedure operatorcoefl similar to the standard REDUCE procedure coefl, but now for linear spaces instead of polynomial rings.

The description of a graded differential geometry package in REDUCE is given. The procedures are useful in the study of supersymmetric equations. The supersymmetric modified KdV equation is discussed as an application.

The structure of the nonlinear Schrödinger prolongation algebra, introduced by Estabrook and Wahlquist, is explicitly determined. It is proved that this Lie algebra is isomorphic with the direct product H× (A1 ⊗ C[t]), where H is a three-dimensional commutative Lie algebra.

For a coupled system of KdV equations the prolongation Lie algebra is explicitly determined. It turns out to be of Kac-Moody type.

Symmetries and higher-order or generalised symmetries for the SKdV equation are constructed. Moreover by the introduction of graded potentials a non-local generalised symmetry is obtained, leading to the recursion operator for symmetries in a straightforward way, without making use of the bi-Hamiltonian structure.

The main aim of this paper is to contribute to the automatic calculations in differential geometry and its applications, with emphasis on the prolongation theory of Estabrook and Wahlquist, and the calculation of invariance groups of exterior differential systems. A large number of worked examples have been included in the text to demonstrate the concrete manipulations in practice. In the appendix, a list of programs discussed in the paper is added.

We present and discuss a three-parameter class of exact solutions to the localized-induction-approximation equations. These are one-soliton excitations (Backlund transforms) of the circular vortex motion. The corresponding generic vortex filament (of infinite or finite length) remains in the interior of the sphere (or torus) moving with a constant velocity. The above solutions generate a new class of exact solutions to the classical one-dimensional continuous Heisenberg ferromagnet model.