In this paper, the author studies all the elliptic integrable systems, in the sense of C, that is to say, the family of all the \(m\)-th elliptic integrable systems associated to a \(k^\prime\)-symmetric space \(N=G/G_0\). The author describes the geometry behind this family of integrable systems for which we know how to construct (at least locally) all the solutions. The introduction gives an overview of all the main results, as well as some related subjects and works, and some additional motivations.

Table of Contents

• Introduction
• Notation, conventions and general definitions
• Invariant connections on reductive homogeneous spaces
• \(m\)-th elliptic integrable system associated to a \(k^\prime\)-symmetric space
• Finite order isometries and twistor spaces
• Vertically harmonic maps and harmonic sections of submersions
• Generalized harmonic maps
• Generalized harmonic maps into \(f\)-manifolds
• Generalized harmonic maps into reductive homogeneous spaces
• Appendix
• Bibliography
• Index
• List of symbols