A new nonlinear dynamical system that leads to eigenvalues (Q1185123)

A new dymamical system of Lax type \(dL(t)/dt=[L(t),[J(L(t)),L(t)]]\) is introduced, where \(L(t)\) is a function with value in the Lie algebra \(\text{su}(n)\). Set \(L^ 0=L(0)\) and define an isospectral manifold \({\mathcal I}_{L^ 0}\) on \(\text{su}(n)\) by the adjoint orbit of compact Lie group \(\text{SU}(n)\) on \(\text{su}(n)\) through \(L^ 0\), i.e. \({\mathcal I}_{L^ 0}=\{g^{-1}L^ 0g\mid g\in \text{SU}(n)\}\). Let \(\eta\) be the Cartan \text{su}balgebra of the \(\text{su}(n)\). The paper shows that the set of stable fixed points of the dynamical system lies in \({\mathcal I}_{L^ 0}\cap \eta\).