On an isospectral Lie–Poisson system and its Lie algebra

In this paper we analyse the matrix differential system $X_0 = [N,X^2]$, where $N$ is skew-symmetric and $X(0)$ is symmetric. We prove that it is isospectral and that it is endowed with a Poisson structure, and discuss its invariants and Casimirs. Formulation of the Poisson problem in a Lie–Poisson setting, as a flow on a dual of a Lie algebra, requires a computation of its faithful representation. Although the existence of a faithful representation is assured by the Ado theorem and a symbolic algorithm, due to de Graaf, exists for general computation of faithful representations of Lie algebras, the practical problem of forming a ‘tight’ representation, convenient for subsequent analytic and numerical work, belongs to numerical algebra. We solve it for the Poisson structure corresponding to the equation $X_0 = [N,X^2]$.