The height invariant of a four-parameter semitoric system with two focus-focus singularities

Semitoric systems are a special class of completely integrable systems with two degrees of freedom that have been symplectically classified by Pelayo and Vu Ngoc about a decade ago in terms of five symplectic invariants. If a semitoric system has several focus-focus singularities, then some of these invariants have multiple components, one for each focus-focus singularity. Their computation is not at all evident, especially in multi-parameter families. In this paper, we consider a four-parameter family of semitoric systems with two focus-focus singularities. In particular, apart from the polygon invariant, we compute the so-called height invariant. Moreover, we show that the two components of this invariant encode the symmetries of the system in an intricate way.


Introduction
In the last decades, various efforts have been made towards the construction of classifications within the theory of completely integrable dynamical systems. These classifications are based on invariants that capture various aspects of a system with respect to different notions of equivalence. They are useful for two main reasons: they give an overview of all possible systems within a certain class and allow us to distinguish between nonequivalent systems. If we restrict ourselves to classifications of symplectic type, important accomplishments are the classification of toric systems, due to Delzant [10], Atiyah [6] and Guillemin & Sternberg [15] and the classification of semitoric systems, due to Pelayo & Vũ Ngo . c [26,27] and recently extended by Palmer et al. [23]. Another significant result in this line is the symplectic classification of completely integrable systems using characteristic classes, introduced by Zung [36].
Semitoric systems are a class of dynamical systems defined on connected four-dimensional symplectic manifolds, introduced by Vũ Ngo . c [32]. They are integrable systems, so they have two conserved quantities, one of which is a proper map that induces an effective circle action. Moreover, all singularities are required to be non-degenerate and must not have hyperbolic components. From a topological point of view, these systems can be described using the theory of singular Lagrangian fibrations, cf. Bolsinov & Fomenko [7].
From the symplectic point of view, one of the motivations to study semitoric systems comes from the analysis of systems with monodromy in the quantum physics and chemistry literature, see for example Child et. al. [8], Sadovksii & Zhilinskii [29] for a theoretical approach and Assémat et. al. [5], Fitch et. al. [14], Winnewisser et. al. [35] for experimental studies.
In this setting, one has the joint spectrum of a set of unknown quantum operators and wants to recover information about the system. An overview of the possible candidate systems can be obtained by means of a classification. Since classical systems are generally easier to understand, one can make use of Bohr's correspondence principle or Zauberstab and focus on constructing a classification for classical systems. However, in order for the results to be valid after quantisation, it is important that this classsification preserves the symplectic structure, cf. Pelayo [24] for more details on this approach.
Two foundational examples of the semitoric systems theory are the coupled spin-oscillator and the coupled angular momenta. The first one is a particular case of the Jaynes-Cummings [17] model from quantum optics and it consists of the coupling of a classical spin on the two-sphere S 2 with a harmonic oscillator on the plane R 2 , see e.g. Pelayo & Vũ Ngo . c [28]. The second one is the classical version of the addition of two quantum angular momenta, defined on the product of two copies of S 2 . It models, for example, the reduced Hamiltonian of a hydrogen-like atom in the presence of parallel electric and magnetic fields, cf. Sadovskii et al. [30]. In the last years, several other examples of semitoric systems have been discovered: Hohloch & Palmer [16] introduced a family with two focus-focus points, Le Floch & Palmer [18] proved the existence of examples in all Hirzebruch surfaces and De Meulenaere & Hohloch [9] proposed a system with four focus-focus points that has double pinched focus-focus fibres for a certain value of the parameter.
The classification of semitoric systems is based on five symplectic invariants: the number of focus-focus points, the polygon invariant, the height invariant, the Taylor series invariant and the twisting index invariant.
The survey article by Alonso & Hohloch [3] gives an overview of the state of the art concerning examples and computations of invariants reached in 2019. Note that the computation of these invariants is far from trivial, especially if the aim is to make a general calculation of the invariants for a whole family of systems depending on several parameters, instead of for only one explicitly given, concrete system. So far, the full list of invariants has only been computed for the two foundational examples. The computation of the invariants in these two cases is based on the use of the properties of elliptic integrals. In the case of the coupled spin-oscillator, it was initiated by Pelayo & Vũ Ngo . c [28] and completed by Alonso & Dullin & Hohloch [1]. In this case, two parameters are taken into account, but the dependence is quite simple. For the coupled angular momenta, it was initiated by Le Floch & Pelayo [19] and completed by Alonso & Dullin & Hohloch [2]. In this case, the dependence is of three parameters and significantly more involved.
Expressing the invariants as a function of the parameters of the system is important because, besides the quantitative results, it also allows for qualitative considerations. For instance, one can compare the roles played by geometric parameters, i.e. those related to the symplectic manifold, and by coupling parameters, i.e. those only appearing in the momentum map. In case some parameters also affect the type of singularities, for example making focus-focus singularities appear and disappear, one can also see what happens to the invariants as the critical values of the parameters are approached.
In both foundational examples, the invariants display the symmetries of the systems. Moreover, for the coupled angular momenta, the terms of the Taylor series invariant go to infinity as the coupling parameter approaches the critical values. However, a limitation of these examples is that the number of focus-focus points is at most one. Semitoric systems with more than one focus-focus point are interesting because, in this case, the symplectic invariants have multiple components, one for each focus-focus point. So the different components can (and should) be compared with each other. In particular, it is interesting to see how the different components depend on the parameters of the system and how they reflect the possible symmetries of the system.
Note that the presence of multiple focus-focus points increases the complexity of the computations significantly. So far, the only results in this direction are the computation of the polygon invariant and the height invariant of two families of systems with a relatively simple dependence on two parameters, cf. Le Floch & Palmer [18]. There is, in general, a certain trade-off between, on the one hand, the qualitative richness of having the invariants expressed as functions of several parameters and, on the other hand, the feasability of their computations.
In the present paper we choose the former option, i.e., focusing on dependence on multiple parameters. We managed to compute the number of focus-focus points invariant, the polygon invariant, and the height invariant, but, given the number of parameters, the computational complexity of the Taylor series invariant and the twisting index invariant is beyond current computational methods and resources.
where ω S 2 is the standard symplectic form of the unit sphere and 0 < R 1 < R 2 or 0 < R 2 < R 1 . Given (x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ) Cartesian coordinates in M and s 1 , s 2 ∈ [0, 1], we consider the integrable system (M, ω, F ), where F := (L, H) is defined by It is a family of semitoric systems that can have up to two focus-focus singularities and depends on four parameters in total, two geometric parameters R 1 , R 2 > 0, R 1 = R 2 and two coupling parameters Our first result is the computation of the number of focus-focus singularities:

Theorem 1. The number of focus-focus points invariant of system
If E = 0, the system fails to be semitoric.
Theorem 1 is a reformulation of Theorem 15 and Corollary 16 stated later in the paper. The number of focus-focus points invariant is illustrated in Figure 6. The image of the momentum map of system (1) is plotted in Figure 5 which is the starting point for the computation of the polygon invariant. Figure 7.

Theorem 2. The polygon invariant is computed in Theorem 21 and plotted in
Whenever n FF = 2, the height invariant is defined and has two components. Their explicit computation is our main result: For the values of (s 1 , s 2 ) for which the system (1) has two focus-focus singularities, the height invariant h := (h 1 , h 2 ) is given by The height invariant is plotted in Figure 1 and the coefficients γ A γ B , γ C and γ D are explicitly stated in Proposition 4 below. The coefficients encode the dependence of the height invariant on the various parameters. This dependence is polynomial, except for some radicals.

Proposition 4.
The coefficients γ A γ B , γ C and γ D of Theorem 3 are given by Theorem 3 and Proposition 4 are restated and proven as Theorem 22 later in the paper. Corollary 5 reappears as Corollary 23 at the very end of the paper.
All computations in this paper were verified with Mathematica.

Structure of the paper
This paper is structured as follows. In section 2, we briefly summarise the definition of simple semitoric systems and the classification in terms of symplectic invariants. In section 3, we introduce our family of semitoric systems and compute the number of focusfocus points and the polygon invariant. Section 4 is devoted to the computation of the height invariant associated to both focus-focus singularities.

Figures
All figures have been made with Mathematica. Figure 2 has also been edited with Inkscape.

Definition 6.
A semitoric system is a completely integrable system (M, ω, F ) with two degrees of freedom, where F := (L, H) : M → R 2 is smooth and the following conditions are satisfied: 1) All singularities are non-degenerate and have no hyperbolic components.
2) The map L induces an effective S 1 -action on M with a 2π-periodic flow.
3) L is proper, i.e. the preimage of a compact set by L is compact again.
Moreover, if the following condition is satisfied, the semitoric system is said to be simple: In each level set of L there is at most one singularity of focus-focus type.
In the present work we will only consider simple semitoric systems. In the context of semitoric systems, we have two degrees of freedom and we exclude hyperbolic components, so the rank of DF can only be 0 or 1. Singularities of rank 0 can either be of focus-focus type or of elliptic-elliptic type, i.e. having two elliptic components. Singularities of rank 1 must necessarily have a regular and an elliptic component, so they are called ellipticregular singularities.

The singular Lagrangian fibration
The momentum map F := (L, H) of a semitoric system induces a two-dimensional singular Lagrangian fibration on M . The base of this fibration is the image B := F (M ), which is a contractible subset of R 2 . Since L is proper, we know that all fibres of F are compact. Vũ Ngo . c [32] showed that the fibres of semitoric systems are connected and he also characterised the structure of the fibration over B. The boundary ∂B ⊂ B consists of all elliptic-elliptic and elliptic-regular critical values. The former are located in the vertices, while the latter always come in one-parameter families and form the edges. The set B F F of focus-focus critical values is finite and lies in the interior of B. The regular fibres are thus mapped to B reg :=B\B F F . Singular fibres are those containing a singularity. Elliptic-elliptic singularities constitute always their own fibre. Elliptic-regular fibres are homeomorphic to a circle. Fibres containing a focus-focus singularity are homeomorphic to a pinched torus, see Figure 2. If simplicity is not assumed (cf. Definition 6), then fibres containing more than one focusfocus singularity, homeomorphic to a multi-pinched torus, are also possible. This situation has been studied by Pelayo & Tang [25] and Palmer et al. [23].
Regular fibres are those containing no singularities. According to the action-angle theorem by Liouville & Arnold & Mineur [4], regular fibres are homeomorphic to the two-torus T 2 . More precisely, for each regular value c ∈ B reg we can find a neighbourhood U of c and V ⊂ R 2 of the origin such that varying smoothly with c. Then the action coordinates (I 1 , I 2 ) on V are given by the expression where is any semiglobal primitive of the symplectic form, d = ω.
Since the Hamiltonian flow of L induces a global circular action on M , we can take γ 1 (c) to be the orbit of L. This way, we will have I 1 (c) = L (c). Different choices of γ 2 (c) belonging to different homology classes will result in different values of I 2 (c), so there is an integer degree of freedom in the definition of this action coordinate. Figure 2: Example fibration of a semitoric system, corresponding to the coupled angular momenta for t = 1/2 and R 2 > R 1 . The fibration has three elliptic-elliptic fibres, one focus-focus fibre and three 1-parameter families of elliptic-regular fibres. On B reg the fibres are regular 2-tori.

The polygon and height invariants
Vũ Ngo . c [32] used the action coordinates to define the so-called cartographic homeomorphism as follows. Let n FF ∈ Z be the number of focus-focus points and c 1 , ..., c n FF ∈ B their critical values. For each r = 1, ..., n FF , pick a sign choice r ∈ {−1, 1} Z 2 and consider the half-line b r r ⊂ B that starts in c r and extends upwards if r = +1 and downwards if Then for any set of choices = ( 1 , ..., n FF ) there exists a map f := f : B → R 2 that is a homeomorphism onto its image ∆ := f (M ), it preserves the first coordinate, i.e. f (l, h) = (l, f (2) (l, h)) and f | B\b is a diffeomorphism onto its image. This process is illustrated in Figure 3. The map f is constructed by extending the coordinates (L, I 2 ) defined by equation (2). The non-smoothness along the segments b r is a consequence of the monodromy induced by the presence of focus-focus singularities, an obstruction to globally-defined action-angle coordinates studied, among others, by Nekhoroshev [22] and Duistermaat [11].
The image ∆ ⊂ R 2 of the cartographic homeomorphism is a convex rational polygon, which is compact if and only if M is compact. Since the definition of the action I 2 is not unique, neither is ∆. There is a Z-action that relates all possible choices of I 2 . Besides that, there is a (Z 2 ) n FF -action of sign choices that also acts on f .
where π 2 : R 2 → R is the canonical projection onto the second coordinate. This quantity is independent of the choice of map f . The height h r can also be interpreted as the symplectic volume of the submanifold Y − r := {p ∈ M | L(p) = L(m r ) and H(p) < H(m r )}, that is, the real volume of Y − r divided by 2π, where m r ∈ M is the focus-focus singularity corresponding to c r . Definition 8. The height invariant associated to the simple semitoric system (M, ω, F ) is the n FF -tuple h = (h 1 , ..., h n FF ). It is independent of the choice of cartographic homeomorphism f .

The other invariants and the symplectic classification
The remaining two invariants are related to the structure of the action I 2 as we approach focus-focus singularities. Fix r ∈ {1, ..., n FF }. In this paper, we do not work with these two invariants, but for sake of completeness, we review them quickly. More details can be found in Pelayo & Vũ Ngo . c [26,27]. In a neighbourhood of the singular fibre containing m r , Vũ Ngo . c [31] proved that the action I 2 can be written as where w := l + ij, i is the imaginary unit, l is the value of L − L(m r ), and j is the value of the second Eliasson function around m r . The function S r (w) is a smooth function that can be understood as a desingularised action. Different choices of I 2 change S r by a multiple of 2πl, so we can fix a choice I 2,r of I 2 in this neighbourhood by imposing 0 ≤ ∂ l S r (0) < 2π. If we denote its Taylor series by S ∞ r , then the n FF -tuple S ∞ = (S ∞ 1 , ..., S ∞ n FF ) is the Taylor series invariant. In §4.3 we show that h r can also be related to I 2,r (0).
Fix now a polygon ∆ and its corresponding homeomorphism f . Then, for each r = 1, ..., n FF , the values of f (2) around c r will differ from those of I 2,r by a multiple κ r ∈ Z of 2πl. The n FF -tuple κ = (κ 1 , ..., κ n FF ) depends on the choice of f . The equivalence class of κ under the (Z 2 ) n FF × Z-action that acts on f determines the twisting index invariant.
Pelayo & Vũ Ngo . c [26,27] give a classification of simple semitoric systems up to isomorphism using the number of focus-focus points n FF and the other four invariants introduced in this section.

A symmetric family with two focus-focus points
The parameters R 1 , R 2 are called geometric parameters, because they are related to the symplectic manifold. The parameters s 1 , s 2 ∈ [0, 1] are the coupling parameters of the system. For now, we will assume that R 2 > R 1 . The function L represents the sum of the height functions on both spheres and its Hamiltonian vector field corresponds to a simultaneous rotation of both spheres around the vertical axis. The function H corresponds to an interpolation among rotations around the vertical axis on the first sphere, the second sphere and the relative polar angle between the two position vectors, see Figure 4.
The four extreme cases considered in the proof of Proposition 12 are actually of toric type, that is, toric up to a diffeomorphism on the base, cf. Vũ Ngo . c [32, Definition 2.1].
In particular, all their flows are periodic. This is because the flow of H = ±z i , i = 1, 2 corresponds to rotations around the vertical axis in the i-th sphere.
In Figure 5 we can see the evolution of the image of the momentum map (L, H) as we move the coupling parameters s 1 , s 2 . The extreme cases correspond to the images on the four corners.

The number of focus-focus points
The first symplectic invariant that we compute is the number of focus-focus points.
The number of focus-focus singularities as a function of the system parameters s 1 , s 2 is illustrated in Figure 6.
Finally, there are no local extrema in the interior of the region,D just grows indefinitely as R → ∞. We conclude thus thatD is positive in all the region. This means that the discriminant D is positive, too, and therefore the singularities are non-degenerate and of elliptic-elliptic type.
• Case N × N and S × S for s 1 = 1 2 : In this case the discriminant D vanishes, so we may compute instead the characteristic polynomial of A L + A H = Ω −1 (D 2 L + D 2 H), which is It is also quadratic in Y = X 2 and has discriminant 4(1 + 8s 2 + 8s 2 2 − 32s 2 3 + 16s 2 4 ) Therefore, the singularities are non-degenerate and of elliptic-elliptic type.
• Case N × S and S × N for s 1 = 1 2 : In this case the discriminant becomes: Since s 1 = 1 2 , the first three factors are always positive. The fourth factor, which we denote by E, determines a closed curve γ. Outside the curve, the discriminant is positive and therefore the singularities are of elliptic-elliptic type. Inside the curve, the discriminant is negative and therefore the singularities N × S and S × N are of focus-focus type.
• Case N × S and S × N for s 1 = 1 2 : In this case the discriminant D vanishes, so we can compute instead the character- which is also quadratic in Y = X 2 and has discriminant − 4(1 + 8s 2 + 8s 2 2 − 32s 2 3 + 16s 2 4 ) Therefore, the singularities are non-degenerate and of focus-focus type.
We now look at the singularities of rank 1.
Proof. We make use of [16,Proposition 3.14], that provides a criterion for the singularities of rank 1. More specifically, let l be the fixed value of the function L, i.e., l := R 1 z 1 + R 2 z 2 and set ϑ := θ 1 −θ 2 , where θ 1 , θ 2 are the polar angles on S 2 ×S 2 . We consider the symplectic reduction of the system (4) on the level L −1 (l). Note that from [16, Lemma 3.10], rank 1 singularities always satisfy z 1 , z 2 = ±1. Define which is the content of the square root after the transformation of H in the reduced coordinates (ϑ, z 1 ), cf. equation (7) in the next section. Then the fixed points of rank 1 are non-degenerate and of elliptic-regular type if In our case, t 4 = 0, so the left hand side vanishes (cf. Remark 11). The right hand side is The numerator lies always between −(α + β) 2 and −(α − β) 2 , where α := R 1 (1 − z 1 2 ) and β := R 2 (1 − z 2 2 ), because −1 < z 1 z 2 < 1, so it is always negative and the denominator is always positive. Thus, the right hand side of (6) is always negative and the criterion [16, Proposition 3.14] can be applied. We conclude that all singularities of rank 1 are non-degenerate.

Theorem 15. The system (4) is semitoric for almost any choice of coupling parameters
It only fails to be semitoric in the piecewise smooth curve defined by E = 0, where E is defined in equation (5). At this curve, the singularities N × S and S × N become degenerate.
Proof. Immediate from Propositions 12, 13 and 14. The curve E = 0 is the border of the coloured region in Figure 6.  Proposition 17. We define the following transformations, acting on the base manifold (M, ω) and the system parameters (R 1 , R 2 , s 1 , s 2 ): They preserve the symplectic form, Ψ * i ω = ω, and act on the system (4) as Proof. Simple substitution in equation (4).

The polygon invariant
The polygon invariant of the system (4) can be obtained from the isotropic weights of the function L by making use of the following theorem, that relates the slopes of the polygon to the derivative of the Duistermaat-Heckman function ρ L (l), that is, the symplectic volume of the reduced space L −1 (l)/S 1 .
where a, b are the corresponding isotropic weights.
Since the phase space (M, ω) and the function L coincide with that of the coupled angular momenta, we can use the isotropic weights computed by Le Floch & Pelayo [19]. We obtain the following result: The polygon invariant of the system (4) is determined by the following cases: • If n FF = 2, the polygon invariant is the ((Z 2 ) 2 × Z)-orbit generated by any of the polygons represented in Figure 7. Proof. We distinguish between two situations: • Case n FF = 2: The polygon invariant consists of the quotient of a ((Z 2 ) 2 × Z)-orbit of a polygon by the ((Z 2 ) 2 × Z)-action, where the action of (Z 2 ) 2 comes from the choice of signs = ( 1 , 2 ), one per focus-focus singularity.
In this case, the polygons coincide, in principle, with those of the coupled angular momenta, namely Figures 7b and 7d but, since we have an additional focus-focus singularity, we also have an additional sign choice, corresponding to the cutting direction on the second singularity. This means that we have to add two more polygons, that is, the ones that coincide with 7b and 7d on the left of the image of the second focus-focus singularity and change the slope by one on its right, cf. Figures 7a and 7c.
• Case n FF = 0: In this case we do not have any sign choice, so the polygon invariant is the quotient of a Z-orbit by the Z-action. If n FF = 0 and (s 1 , s 2 ) lies in the same connected component as the point (0, 0) or (1,1), by looking at Figure 5 and comparing it with Figure 7, we identify that the right polygon is Figure 7b.
If n FF = 0 and (s 1 , s 2 ) lies in the same connected component as the point (1, 0) or (0, 1), then the comparison gives the polygon in Figure 7c.

The height invariant
We compute now the height invariant of the system (4) for the values of s 1 , s 2 for which there are two focus-focus singularities. We start by rewriting the system in a more convenient form. Take cylindrical coordinates (θ 1 , z 1 , θ 2 , z 2 ) on M , so that the system (4) becomes with the symplectic form ω = −(R 1 dθ 1 ∧dz 1 +R 2 dθ 2 ∧dz 2 ). We now perform the following affine transformation, which leads to L( q 1 ,p 1 , q 2 ,p 2 ) =p 1 and with symplectic form ω = d q 1 ∧ dp 1 + d q 2 ∧ dp 2 . In these coordinates, we see that the function H is independent of q 1 . We simplify now our notation by scaling some of the variables and functions by a factor of R 1 . We use caligraphic letters to refer to scaled functions and standard letters for unscaled functions: This way we can express our problem entirely in terms of R := R 2 R 1 . We will do now singular symplectic reduction by the S 1 -action generated by L. We will use two different sets of notations, one adapted to the singularity N × S and the other to the singularity S × N .

Reduced system for the singularity N × S
We start by doing symplectic reduction on the level L = l + (1 − R), which is singular in l = 0, since the focus-focus point N × S lies on the level set l = 0. Expressed in coordinates (q 2 , p 2 ), we obtain the reduced Hamiltonian From this equation we see that the physical region, i.e. the domain of definition of l and p 2 is given by l ∈ [−2, 2R] and p 2 satisfying the inequalities p 2 ≥ 0, p 2 ≥ l, p 2 ≤ 2R and p 2 ≤ l + 2. The region is depicted in Figure 8. For simplicity, we write We now define the polynomial which is defined in such a way that the singularity N × S lies precisely on (l, h) = (0, 0). The polynomial is of degree 4 in p 2 .

Reduced system for the singularity S × N
We focus now on the singularity S × N , so we do symplectic reduction on the level L = l + (1 − R), which is singular in l = 0 since the focus-focus point S × N lies on the level set l = 0. We also change the coordinates (q 2 , p 2 ) to in order to create a similar notation to the one of §4.1 while preserving the symplectic form. In these new coordinates, the reduced Hamiltonian becomes The physical region in this case will be given by l ∈ [−2R, 2] and p 2 satisfying the inequalities p 2 ≥ 0, p 2 ≥ −l, p 2 ≤ 2R and p 2 ≤ −l + 2. The region is depicted in Figure  9. We write We now define the polynomial in such a way that the singularity S × N lies precisely on (l, h) = (0, 0). The polynomial is also of degree 4 in p 2 .

Computation of the height invariant
Now that we have the two reduced models, the next step is to compute the height invariant h = (h 1 , h 2 ) of the system (4). We recall from §2 that the height h r associated to the focus-focus singularity m r ∈ M is the symplectic volume of As suggested by Proposition 17, we are dealing with a very symmetric situation. In particular, the transformation s 1 → 1 − s 1 brings the situation of the singularity N × S to the situation of the singularity S × N and vice versa. In Figure 11, we can see a plot of this volume for the singularity N × S, so r = 1 and, in Figure 12, we can see the same for the case r = 2.
Theorem 22. The height invariant h := (h 1 , h 2 ) associated to the system (4) for the values of (s 1 , s 2 ) in which it has two focus-focus singularities is given by where R := R 2 R 1 , u is the Heaviside step function and The invariant is represented in Figure 10. The coefficients γ A , γ B , γ C and γ D are given by Proof. We start by focussing first on the singularity N × S, i.e. r = 1. We know that (L, H)(m 1 ) = (1 − R, (1 − 2s 1 )(1 − 2s 2 )) and we want to compute the symplectic volume of }, which is the area represented in Figure 11, divided by 2π. In the notation of §4.1, the singularity lies at (l, h) = (0, 0). This means that and The phase space is given by −π ≤ q 2 ≤ π and 0 ≤ p 2 ≤ 2. The roots of P N S 0 (p 2 ) are ζ 1 = ζ 2 = 0 and the physical region lies between ζ 2 and ζ 3 . There are two trivial cases, s 1 = 1 2 and s 2 = R R+1 . In both cases, P N S 0 (p 2 ) := B N S 0 (p 2 ) and therefore the roots are ζ 1 = ζ 2 = 0, ζ 3 = 2 and ζ 4 = 2R. These trivial cases form the border between two different behaviours, i.e., we distinguish the following situations: If we look at Figure 11, we see that cases I and V, bottom-left and top-right respectively, are connected from above. Case III, in the centre, corresponds to the trivial transition situation. Cases II and IV, top-left and bottom-right respectively, are not connected from above. Therefore, for cases I and V we write and for the cases II and IV we write -π -π For the trivial case III we have h 1 = 1 2π 1 2 4π = 1.
As we did in for the general action integral in §4.1, we integrate by parts to compute this integral. For the cases I and V we obtain Q(p 2 ) dp 2 .
Here it is important to observe that the polynomial Q(p 2 ) is of degree 2 in p 2 , so the integral (9) is not elliptic but can be solved explicitly in terms of elementary functions. We will need the following definite integrals In the last step we have used the identity arctan(z) = i 2 (log(1 − iz) − log(1 + iz)) with .
By substituting N A (α, β, γ) and N B (α, β, γ, δ) we obtain the desired result. The proof for the singularity S × N is completely analogous but taking into account that the cases II and V should be exchanged and the same for the cases I and IV. Note that the condition R 1 = R 2 results in a non-simple semitoric system, which lies outside the scope of the present work.