About six years ago, semitoric systems on 4-dimensional manifolds were classified by Pelayo & Vũ Ngo . c by means of five invariants. A standard example of such a system is the coupled spin-oscillator on S 2 × R 2 . Calculations of three of the five semitoric invariants of this system (namely the number of focus-focus singularities, the generalised semitoric polygon, and the height invariant) already appeared in the literature, but the so-called twisting index was not yet computed and, of the so-called Taylor series invariant, only the linear terms were known.In the present paper, we complete the list of invariants for the coupled spin-oscillator by calculating higher order terms of the Taylor series invariant and by computing the twisting index. Moreover, we prove that the Taylor series invariant has certain symmetry properties that make the even powers in one of the variables vanish and allow us to show superintegrability of the coupled spin-oscillator on the zero energy level.Semitoric systems, which in addition allow for so-called focus-focus singularities, have arXiv:1712.06402v2 [math.SG]

The coupled angular momenta are a family of completely integrable systems that depend on three parameters and have a compact phase space. They correspond to the classical version of the coupling of two quantum angular momenta and they constitute one of the fundamental examples of so-called semitoric systems. Pelayo & Vũ Ngo . c have given a classification of semitoric systems in terms of five symplectic invariants. Three of these invariants have already been partially calculated in the literature for a certain parameter range, together with the linear terms of the so-called Taylor series invariant for a fixed choice of parameter values.In the present paper we complete the classification by calculating the polygon invariant, the height invariant, the twisting-index invariant, and the higher-order terms of the Taylor series invariant for the whole family of systems. We also analyse the explicit dependence of the coefficients of the Taylor series with respect to the three parameters of the system, in particular near the Hopf bifurcation where the focus-focus point becomes degenerate.

The flower derivative) action, which describes the actual degrees of freedom of a (higher derivative) lheoly of gravity quadratic in the scalar and Ricci curvatures, is found. Key steps are a Legendre transform and a suitable diagonalidon procedure. Some consequences of this insight are outlined.

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