Global Action-Angle Variables for Non-Commutative Integrable Systems

In this paper we analyze the obstructions to the existence of global action-angle variables for regular non-commutative integrable systems (NCI systems) on Poisson manifolds. In contrast with local action-angle variables, which exist as soon as the fibers of the momentum map of such an integrable system are compact, global action-angle variables rarely exist. This fact was first observed and analyzed by Duistermaat in the case of Liouville integrable systems on symplectic manifolds and later by Dazord-Delzant in the case of non-commutative integrable systems on symplectic manifolds. In our more general case where phase space is an arbitrary Poisson manifold, there are more obstructions, as we will show both abstractly and on concrete examples. Our approach makes use of a few new features which we introduce: the action bundle and the action lattice bundle of the NCI system (these bundles are canonically defined) and three foliations (the action, angle and transverse foliation), whose existence is also subject to obstructions, often of a cohomological nature

Lotka–Volterra systems satisfying a strong Painlevé property

We use a strong version of the Painlevé property to discover and characterize a new class of n-dimensional Hamiltonian Lotka–Volterra systems, which turn out to be Liouville integrable as well as superintegrable. These systems are in fact Nambu systems, they posses Lax equations and they can be explicitly integrated in terms of elementary functions. We apply our analysis to systems containing only quadratic nonlinearities of the form aijxixj, i =j, and require that all variables diverge as t−1. We also require that the leading terms depend on n −2free parameters. We thus discover a cocycle relation among the coefficients aijof the equations of motion and by integrating the cocycle equations we show that they are equivalent to the above strong version of the Painlevé property. We also show that these systems remain explicitly solvable even if a linear term bixiis added to the i-th equation, even though this violates the Painlevé property, as logarithmic singularities are introduced in the Laurent solutions, at the first terms following the leading order pole