Classification scheme for two-dimensional Ermakov-type systems and generalizations

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Abstract
We describe a classification scheme for certain pairs of coupled, nonlinear second-order ordinary differential equations, having a first integral which can be obtained by an Ermakov-type elimination procedure. The classification produces a hierarchy of systems having different levels of generality. Different cases within the same level are distinguished by the integrability of a certain one-form, which is related to a nonlinear superposition law between the two equations. The study of these one-forms and the classification are made systematic by using the general reduction theorem of Pfaffian forms. The paper contains illustrations of the way previous examples fit into the classification scheme, and a discussion of its generalization to higher dimensions.