It is well known that twistor constructions can be used to analyse and toobtain solutions to a wide class of integrable systems. In this article weexpress the standard twistor constructions in terms of the concept of anadmissible family of rational curves in certain twistor spaces. Examples of ofsuch families can be obtained as subfamilies of a simple family of rationalcurves using standard operations of algebraic geometry. By examination ofseveral examples, we give evidence that this construction is the basis of theconstruction of many of the most important solitonic and algebraic solutions tovarious integrable differential equations of mathematical physics. This ispresented as evidence for a principal that, in some sense, all soliton-likesolutions should be constructable in this way.
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