Hierarchies of integrable equations associated with hyperelliptic Lie algebras (Q2773147)

The aim of this paper is to extend the ``second'' Lie-theoretical approach to the case of the special hyperelliptic Lie algebras \(\widetilde g_{\mathcal H}\), where \(g\) is equal to \(so(d)\), \(sp(d)\) or \(gl(d)\). The authors show that the compatibility condition of the two Hamiltonian flows on the Lie algebras \(\widetilde g^\pm_{\mathcal H}\) or their quotient algebras also leads to the zero-curvature equations. In such a way they obtain new hierarchies of integrable Hamiltonian equations admitting zero-curvature representations. Moreover they show that the simplest equations of the hierarchy coincide with a kind of multiparametric deformation of the generalized Heisenberg magnet equations, where the parameters of the deformations are the branching points of the hyperelliptic curve.