On degenerate integrable hamiltonian systems

Question

Is there some reference where the existence of local generalized action-angle variables is discussed in some detail for concrete examples of hamiltonian systems of mechanical type?

After Dazord and Delzant, by local generalized action-angle coordinates on a symplectic manifold $(M^{2n},\omega)$ I mean a locally trivial bundle $\pi:M\to P$ with fiber $\mathbb{T}^k$ and having a trivializing atlas whose elements $(U,\phi:\pi^{-1}(U)\to U\times\mathbb{T}^k)$ satisfy the following property:
$\phi_{\ast}\omega=\sum_{i=1}^k dJ_i\wedge \theta_i+\sum_{i=1}^{n-k}dp_i\wedge dp_i$ where $J_1,\ldots,J_k,p_1,\ldots,p_{n-k},q_1,\ldots,q_{n-k}$ are adapted coordinates on $U$ and $\theta_1\ldots,\theta_k$ is a base of invariant $1$-forms on $\mathbb{T}^k$.

Answer 1

If I understand correctly what you are after, a good place to have a look at is this expository article by Fasso'

F. Fasso'. Superintegrable Hamiltonian systems: geometry and perturbations. Acta Appl. Math. 87 (2005),

in which he reviews the case of superintegrable Hamiltonian systems defined by Mischenko and Fomenko (this is the mechanical case that you should be after). Some examples are briefly discussed and there are references to longer expositions (e.g. the Euler-Poinsot top).

Just a final word of warning. In the completely integrable Hamiltonian systems literature, "degenerate" refers to the case when the underlying system is completely integrable a la Liouville, but there are singularities which are not, in some sense, of Morse-Bott type. If I am not mistaken, the case that you discuss here is often referred to as "superintegrable" or "non-commutatively integrable", as defined by Mischenko and Fomenko (and others).