Algebraic integrable systems related to spectral curves with automorphisms (Q478616)

Using a reduction to the Beauville systems, a family of new algebraic completely integrable systems, related to curves with a cyclic automorphism, is obtained. The structure of the paper is as follows: In Section 2 the relation between the Jacobians of two curves which are linked by a ramified cyclic covering of prime order is studied. In Section 3 a space of polynomial matrices of size \(p\) is introduced and its automorphisms of order \(p\) are studied, with particular attention to the fixed point set of such an automorphism. In Section 4, both the space and its fixed point set are related to the corresponding spectral curves, upon using the momentum map and the results of Section 2. Beauville's result also used to describe the fibers of the algebraic completely integrable systems under construction. The Hamiltonian structure of the space of polynomial matrices, its fixed point sets and their quotients (by the adjoint action) are considered in Section 5. In particular a multi-Hamiltonian structure of the newly constructed phase spaces is also obtained. The algebraic integrability this system is proved in Section 6.