Dual pairs in resonances (Q1943034)

Let \(D\) be an open subset of \(\mathbb C^2\) with the symplectic forms \(\omega_{\pm}\) and \(B\) an open subset of \(\mathbb R^3\). The present paper is devoted to dual pairs \((D,\omega _{\pm})\rightarrow B\), \((D,\omega_{\pm})\rightarrow\mathbb R\) associated to \(n:m\) resonances as well as to \(n:-m\) resonances. The Poisson structure on \(B\), which depends on the natural numbers \(n\) and \(m\), is not Lie-Poisson. Instead, its symplectic leaves are the Kummer shapes: bounded surfaces for \(n:m\) resonances and unbounded surfaces for \(n:-m\) resonances.