  In this paper we build dual pairs of Poisson maps \[
\RR\stackrel{R_\pm}{\longleftarrow}(D,\om_\pm)
\stackrel{\Pi_\pm}{\longrightarrow} B \] associated to $n:m$ resonance, as well
as to $n:-m$ resonance. Except for the above mentioned cases $1:\pm1$, these
are not pairs of momentum maps. Here $D$ is an open subset of $\CC^2$ with the
above mentioned symplectic forms $\om_\pm$, and $B$ an open subset of $\RR^3$.
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 $n:m$ resonance, and unbounded surfaces for $n:-m$
resonance
