On the reduction theory of the Nijenhuis operator and its applications to Gel'fand-Dikii equations, Proc.Iutam-Isimm Symposium on Modern Developments in Analitical Mechanics, ATTI ACC.SCIENZE DI TORINO, Suppl.Vol.117

The aim of this paper is twofold: to develop the basic elements of the reduction theory of the Nijenhuis operators, and to explicitly construct the recursion operator of the general Gelfand-Dickii equations. To deal with the first problem, the geometric properties of the Nijenhuis operator are pointed out: they mainly concern a remarkable integrable distribution defined by such operators. The two “reduction theorems” presented in the paper give the conditions allowing to obtain a reduced kernel-free Nijenhuis operator either on any fixed leaf or on the space of the leaves of this distribution, respectively. In the second part of the paper, a remarkable class of matrix Nijenhuis operators is introduced, depending on an arbitrary matrix with constant entries; the recursion operator of the Gelfand-Dickii equations is then obtained as a particular application of the reduction theory to this class of operators.