We study the symplectic geometry of the Jaynes-Cummings-Gaudin model with $n=2m-1$ spins. We show that there are focus-focus singularities of maximal Williamson type $(0,0,m)$. We construct the linearized normal flows in the vicinity of such a point
and show that soliton type solutions extend them globally on the critical torus. This allows us to compute the leading term in the Taylor expansion of the symplectic invariants and the monodromy associated to this singularity.
We study the Kowalevski expansions near singularities of the swinging Atwoods machine. We show that there is a infinite number of mass ratios $M/m$ where such expansions exist with the maximal number of arbitrary constants. These expansions are of th
e so--called weak Painleve type. However, in view of these expansions, it is not possible to distinguish between integrable and non integrable cases.
