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.

About 6 years ago, semitoric systems were classified by Pelayo & Vũ Ngo . c by means of five invariants. Standard examples are the coupled spin oscillator on S 2 × R 2 and coupled angular momenta on S 2 × S 2 , both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a 6-parameter family of integrable systems on S 2 × S 2 and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.where ω S 2 is the standard volume form on the sphere and 0 < R 1 < R 2 are real numbers. For R := (R 1 , R 2 ) and t := (t 1 , t 2 , t 3 , t 4 ) ∈ R 4 define J R , H t : M → R by (1) where (x i , y i , z i ) are Cartesian coordinates on S 2 ⊂ R 3 for i = 1, 2. Then there exist choices of t 1 , t 2 , t 3 , t 4 , R 1 , R 2 such that (M, ω, (J R , H t )) is a semitoric system with exactly two focus-focus points.Theorem 1.1 is restated in more detail in Section 3 as Theorem 3.1. The coupled angular momenta system with coupling parameter t ∈ ]0, 1[ is the special case of Equation (1) with t 1 = t, t 3 = t 4 = 1−t, and t 2 = 0. The coupled angular momenta system describes the rotation of two vectors (with magnitudes R 1 and R 2 ) about the z-axis and has as a second integral a linear combination of the z-component of the first vector and the inner produce of the two vectors, while the system in Equation (1) includes additionally the z-component of the second vector and also

We introduce a new Floer theory associated to a pair consisting of a Cartan hypercontact 3-manifold M and a hyperkähler manifold X . The theory is a based on the gradient flow of the hypersymplectic action functional on the space of maps from M to X . The gradient flow lines satisfy a nonlinear analogue of the Dirac equation. We work out the details of the analysis and compute the Floer homology groups in the case where X is flat. As a corollary we derive an existence theorem for the 3-dimensional perturbed nonlinear Dirac equation. 53D40, 32Q15