Multi-Hamiltonian formulation for a class of degenerate completely integrable systems

Generalizing a construction of P. Vanhaecke, we introduce a large class of degenerate (i.e., associated to a degenerate Poisson bracket) completely integrable systems on (a dense subset of) the space $\R^{2d+n+1}$, called the generalized master systems. It turns out that certain generalized master systems (with different Poisson brackets and different Hamiltonians) determine the same Hamiltonian vector fields (and are therefore different descriptions of the same Hamiltonian system), and that the Poisson brackets of these systems are compatible. Consequently, our class of generalized master systems actually consists of a (smaller) class of completely integrable systems, and our construction yields a multi-Hamiltonian structure for these systems. As an application, we construct a multi-Hamiltonian structure for the so-called master systems introduced by D. Mumford.