Magic Billiards: the Case of Elliptical Boundaries
Vladimir Dragović Milena Radnović



    The YMDB catalog: Young massive detached binaries for the determination of high-precision absolute stellar parametersFull version of Tables 2, 3 and 4 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.
    
    
    Pablo Martín-Ravelo 
    1,2
    Roberto Gamen 
    3,4
    Julia I. Arias 
    1
    André-Nicolas Chené 
    2
    Rodolfo H. Barbá In Memoriam (1962–2021)
    1
    
    Received: 20 June 2024 / Accepted: 05 July 2024
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Dedicated to Academician A. T. Fomenko on the occasion of his 80th anniversary.





§ ABSTRACT 
In this work, we introduce a novel concept of magic billiard games and analyse their properties in the case of elliptical boundaries.
We provide explicit conditions for periodicity in algebro-geometric, analytic, and polynomial forms.
A topological description of those billiards is given using Fomenko graphs.




Keywords: integrable systems, elliptical billiards, topological billiards, Fomenko graphs, periodic trajectories, Cayley's conditions, divisors on elliptic curves, polynomial Pell's equations



MSC2020: 37C83, 37J35, 37J39








§ INTRODUCTION


The elliptic billiard Bir,KT1991,Tab is a notable example of a completely Liouville integrable system. 
Thus, the Liouville-Arnold theorem implies that the phase space of that particular billiard is foliated into invariant Liouville tori <cit.>. 

Fomenko and his school developed a beautiful theory for topological description and classification of integrable systems using what are now known as Fomenko graphs and Fomenko-Zieschang invariants, see Fomenko1987,FZ1991 and, in particular, the book <cit.>, with the fundamentals of that theory, including a large list of well-known integrable systems, such as the integrable cases of rigid body motion and geodesic flows on surfaces.

The use of topological tools in the study of integrable billiards was initiated by the authors in DR2009, see also DR2010, DR2011. 
Further details and applications to other integrable systems can be found in the literature related to billiards Fokicheva2014, R2015, DR2017, VK2018,FV2019, FV2019a, PRK2020, DGR2021, DGR2022,BF2024.
For the applications in the broader theory of Hamiltonian systems with two degrees of freedom see BMF1990,RRK2008,BBM2010.


An important milestone of this theory is the so-called Fomenko conjecture, emphasizing a surprising universality of billiard dynamics. This conjecture is about  realization of topology of Liouville foliaions of smooth and real-analytic integrable Hamiltonian systems by integrable billiards, see e.g. <cit.> and <cit.> and references therein. 


In this paper, we introduce general magic billiards, where after hitting the boundary the particle is magically transported to another point of the boundary and continues motion from there.
A formal definition of such class of systems is given in
Section <ref>.
In Section <ref>, we focus to magic billiards within an ellipse, and among them only to those ones where the equations of motion in elliptic coordinates remain the same as for the standard billiard, which will mean that the obtained system will be integrable.
We provide conditions for periodicity of such systems and give topological description using Fomenko graphs.
We note that one of the cases we consider, so called billiards with slipping was recently introduced and studied by Fomenko and his school FVZ2021,FV2021,VZ2022,Fom2023,Zav2023.
In Section <ref>, we consider magic billiard within elliptic annulus. 
The last Section <ref> contains discussion.





§ DEFINITION OF MAGIC BILLIARDS


In this section, we will introduce a new class of dynamics, where a particle moves along straight segments by constant speed within a given domain in the plane, and when it reaches its boundary, it is transported to another point of the domain boundary from where it continues the motion within the domain.

More formally, we will introduce that dynamics as follows.

Let D be a given domain in the plane, which is bounded by a smooth closed curve. Suppose that φ a continuous bijective mapping of the boundary ∂ D onto itself.
Now, consider the subset T(∂ D) of the tangent fibration of the plane consisting only of the tangent spaces at the points of the curve ∂ D.
Let φ^* be a continuous bijective mapping of T(∂ D) onto itself, satisfying the following:

	
  * π∘φ^*=φ∘π, where π:T(∂ D)→∂ D is the projection to the base points;
	
  * φ^* maps tangent vectors to the boundary ∂ D to tangent vectors to that boundary curve;
	
  * vectors pointing outwards D are mapped to vectors pointing inwards D.





A magic billiard (D,φ,φ^*) is a dynamical system where a particle moves with a unit speed and without constraints in the interior of D, and when it hits the boundary at a point A with velocity v⃗, it will bounce off at point φ(A) with velocity φ^*(v⃗).



Notice that the standard billiard in D belongs to the class of just defined magic billiards.
There, mapping φ is the identity and φ^* is the billiard reflection, i.e. reflection with respect to the direction of the tangent line to the boundary at each point.




Another class of magic billiards are billiards with slipping, see <cit.>.




§ MAGIC BILLIARDS WITHIN AN ELLIPSE



Before starting the analysis of novel examples of magic elliptical billiards, we will review the standard billiard within an ellipse.

Suppose that the ellipse is given by:

    : x^2/a+y^2/b=1, 
      
    a>b>0.

Following classical ideas of Jacobi, one can introduce the elliptic coordinates (λ_1,λ_2), which are, for each given point in the plane, the parameters of an ellipse and a hyperbola from the confocal family:

    _λ : x^2/a-λ+y^2/b-λ=1,

which intersect at that point.
Note that each billiard trajectory within  has a unique caustic _α, which is touching each segment of the trajectory.
The differential equation of the billiard motion is separated in elliptic coordinates:

    dλ_1/√((a-λ_1)(b-λ_1)(α-λ_1))
    +
    dλ_2/√((a-λ_2)(b-λ_2)(α-λ_2))
    =
    0.



We are interested in magic billiards which keep those nice geometric and analytic properties of standard elliptic billiards, so we will focus to the mappings φ and φ^* which preserve the equation (<ref>).

In particular, we note that the elliptic coordinates remain unchanged for φ being one of the following:

	
  * the reflection with respect to one of the axes of the billiard boundary ; or
	
  * the half-turn around the center of .

Notice that such φ is defined on the whole plane, not only on the boundary of the billiard table.
Moreover, because the elliptic coordinates are invariant with respect to reflections with respect to the axes, the equation (<ref>) will also be invariant if φ^* is the composition of the billiard reflection in the point of impact and the differential map dφ.
Thus, for such φ and φ^*, the dynamics of the magic billiard (,φ,φ^*) in elliptic coordinates will be identical to the usual billiard motion.
In particular, each trajectory of such magic billiards will have a unique caustic from the family of conics which are confocal with the boundary .


Note that for such φ, the configuration space can be defined as 𝒟/∼, where 𝒟 is the billiard table, and x∼φ(x), for x∈=∂𝒟.

The phase space is defined, analogously, when the velocity vectors mapped into each other by φ^* are identified.


Next, we will consider each of those mappings separately.
For each of them, we will derive periodicity conditions in three forms: algebro-geometric, analytic, and polynomial.


Since φ and φ^* are involutions, the n-periodicity conditions for even n will be the same as for the standard elliptical billiard.
For odd periods, the conditions will be different, and will be distinct for each case that we consider. 


We will also present Fomenko graphs corresponding to the Liouville foliation of the isoenergy manifold.



 §.§ 
	Elliptical billiard with flipping over long axis


In this system, φ is the reflection with respect to the longer axis.
Examples of its trajectories are depicted in Figure <ref>.	


Periodicity conditions for such trajectories can be obtained by applying the results from <cit.>.
In the following theorem, we represent such conditions in the algebro-geometric form, from which the classical form of Cayley's and in the polynomial form can be derived.


In the billiard table bounded by the ellipse  given by (<ref>), consider the magic billiard with flipping over the long axis.
Consider a trajectory of such billiard with the caustic _β from the confocal family (<ref>).
Such a trajectory
is n-periodic if and only if one of the following conditions is satisfied:
	
		
  * n is even and nQ_0∼ nQ_∞;
		
  * n is odd, _β is hyperbola, and nQ_0∼ nQ_b. 
	
Here Q_0, Q_b, Q_∞ denote the points with coordinates (0,√(abβ)), (b,0), (∞,∞) on the elliptic curve:

    y^2=(a-x)(b-x)(β-x).


	
	Integrating (<ref>) along a trajectory of the magic billiard, we get that the n-periodicity condition will be equivalent to the following divisor condition:
	
    n(Q_0-Q_c_1)+m(Q_c_2-Q_a)∼0

	on the elliptic curve (<ref>).
	Here, we have {c_1,c_2}={β,b} and c_1<c_2, while Q_β, Q_a, denote the points of the curve with coordinates (β,0), (a,0) respectively.
	Note that, on the trajectory, the elliptic coordinates λ_1 and λ_2 belong to the intervals [0,c_1] and [c_2,a] respectively.
	Natural numbers n and m represent the numbers of times each of them traces back and forth its respective interval along the closed trajectory.
	
	Note that the short axis of the ellipse is the degenerate conic _a of the confocal family.
	Since that axis must be crossed even number of times along a closed trajectory, we have that m is even, thus the periodicity condition (<ref>) reduces to n(Q_0-Q_c_1)∼0.
	
	On the other hand, the sum of the number of the flippings over the long axis and the number of times when the billiard particle actually crossed the long axis must also be even.
	That sum equals n+m if _β is an ellipse, and 2n if _β is hyperbola.
	From there, we conclude that all closed trajectories with ellipse as caustic must have even period.
	
Now, the required conditions follow from the fact that c_1=b when 𝒞_β is hyperbola and 2Q_β∼2Q_∞.


Now, analogously as in <cit.>, we can get the conditions of Cayley's type.


A trajectory of the billiard within  with flipping over the long axis is n-periodic if and only if one of the following conditions is satisfied:

	
  * n is even and

    |	[     B_3     B_4       … B_n/2+1;     B_4     B_5       … B_n/2+2;       …; B_n/2+1 B_n/2+2       …   B_n-1 ]|=0;

	
  * n is odd, the caustic is hyperbola, and 
	
    |	[         C_2         C_3           …   C_(n+1)/2;         C_3         C_4           … C_(n+1)/2+1;           …;   C_(n+1)/2 C_(n+1)/2+1           …       C_n-1 ]|=0.

 
	Here, we denoted:

    √((a-x)(b-x)(β-x))=B_0+B_1x+B_2x^2+…,
    	
    √((a-x)(b-x)(β-x))/b-x=C_0+C_1x+C_2x^2+…,

the Taylor expansions around x=0, and β is the parameter of the caustic from (<ref>).	


The periodicity conditions can be equivalently stated in the form of polynomial equations.
Again, for details on how to obtain those equations, refer to <cit.>.
A more general theory connecting polynomial Pell's equations and integrable billiards in arbitrary dimension is given in <cit.>.

The trajectories of the magic billiard with flipping over the long axis with caustic _β are n-periodic if and only if there exists a pair of real polynomials p_d_1, q_d_2 of degrees d_1, d_2 respectively, and satisfying the following:
	
		
  (a) if n=2m is even,  then d_1=m, d_2=m-2, and
		
    p_m^2(s)
    		-
    		s(s-1/a)(s-1/b)(s-1/β)
    		q_m-2^2(s)=1;

		
  (b) if n=2m+1 is odd, then d_1=m, d_2=m-1, and
		
    (s-1/b)p_m^2(s)
    		-
    		s(s-1/a)(s-1/β)q_m-1^2(s)=-1.

		


We conclude this section by a topological description of the system.


The rough Liouville equivalence class of an isoenergy manifold in the phase space of the magic billiard with flipping over the long axis is given by the Fomenko graph in Figure <ref>.



The billiard table consists of the boundary ellipse  and its interior. 
According to Remark <ref>, the configuration space is obtained when the points of  which are symmetric to each other with respect to the long axis are identified.
Thus, the configuration space is homeomorphic to a sphere, where the boundary of the billiard table is represented by an arc.

Notice that the trajectories of our system do not depend on the velocity of the particle, thus all isoenergy manifolds will be equivalent to each other.
Now, we will fix an isoenergy manifold.

Any level set corresponding to the trajectories having a fixed ellipse as caustic consists of a single Liouville torus.

The level set corresponding to the trajectories having a fixed hyperbola as caustic consists of two Liouville tori: one torus contains the trajectories where motion along each segment is downwards (as shown in the lefthandside of Figure <ref>), and on the other torus the motion is upwards.

The level set corresponding to the caustic =_0 consists of the limit motion back and forth along the boundary.
This is a single closed trajectory, thus corresponding to the Fomenko atom 𝐀.

The level set with the caustic _a corresponds to the motion along the short axis of the ellipse .
When the particle hits the boundary, it is magically flipped to the opposite point on the axis.
Thus, there are two closed trajectories there: the downwards one and the upwards one.
Each of those trajectories corresponds to one Fomenko atom of type 𝐀.

The level set _b consists of the trajectories that contain the foci of the ellipse .
Exactly one of those trajectories is closed, corresponding to horizontal motion along the long axis.
All other trajectories alternately pass through the one and the other focus, and on each such trajectory, because of the flipping along the long axis, the motion is either upwards along each segment or downwards along each segment.
Thus, there are two separatrices on that level set, so it is represented by the Fomenko atom 𝐁.




 §.§ Elliptic billiard with flipping over short axis


In this case, with φ is the reflection with respect to the short axis.
	Examples of its trajectories are depicted in Figure <ref>.


Now, we derive the algebro-geometric condition for periodicity of the such magic billiard.





In the billiard table bounded by the ellipse  given by (<ref>), consider the magic billiard with flipping over the short axis.
Consider a trajectory of such billiard with the caustic _β from the confocal family (<ref>).
Such a trajectory
is n-periodic if and only if n is even and nQ_0∼ nQ_∞.
	
Here Q_0 and Q_∞ are as in Theorem <ref>.


We will use notation as in Theorem <ref> and its proof.
In the same way as explained there, we can derive the divisor condition (<ref>).
	
	In addition to that condition, we need the following:
	
		
  * the number of times that the particle crossed the long axis must be even: that number equals m if the caustic is ellipse, or n if it is hyperbola;
		
  * the sum n+m must be even, since that is the sum of the number of the flippings over the short axis and the number of times when the billiard particle actually crossed that axis.
	
	In any case, we get that both m and n must be even, so the conditions for closure reduce to n(Q_0-Q_c_1)∼0, which is equivalent to the stated relation.



It is interesting to note that in this case there are no odd-periodic trajectories.
Notice that, as announced in Remark <ref>, the conditions for trajectories of even period are the same as in the case of flipping over the long axis, see Theorem <ref>, and as in the case of standard billiard, see <cit.>*Theorem 2.


As a consequence, the analytic and polynomial conditions can be obtained as follows.


In the billiard table bounded by the ellipse  given by (<ref>), consider the magic billiard with flipping over the short axis.
Consider a trajectory of such billiard with the caustic _β from the confocal family (<ref>).
Such a trajectory
is n-periodic if and only if n is even and the following equivalent conditions are true:

	
  * relation (<ref>) is satisfied;
	
  * there are polynomials satisfying the polynomial equation (<ref>).
 


We use Fomenko graph to give a topological description of the system.


The rough Liouville equivalence class of an isoenergy manifold in the phase space of the magic billiard with flipping over the long axis is given by the Fomenko graph in Figure <ref>.



Similarly as in the proof of Theorem <ref>, the configuration space is homeomorphic to a sphere, where the boundary of the billiard table is represented by an arc.
	
In a fixed isoenergy manifold, any level set corresponding to the trajectories having a fixed non-degenerate conic as caustic consists of a single Liouville torus.
		
The level set corresponding to the caustic =_0  corresponds to the Fomenko atom 𝐀, similarly as in the proof of Theorem <ref>.
	
The level set with the caustic _a corresponds to the motion back and forth along the short axis of the ellipse , thus it corresponds to the Fomenko atom of type 𝐀.
	
The level set with the caustic _b consists of the trajectories that contain the foci of the ellipse .
Two of those trajectories is closed. 
One closed trajectory corresponds to motion to the right along the long axis: when the particle hits the boundary, it is magically flipped to the other endpoint of the diameter, from where it continues its motion to the right.
The second closed trajectory corresponds to the motion to the left along the long axis.

For any other trajectory on that level set, each segment will pass through the same focus.
	Thus, there are two separatrices on that level set, so it is represented by the Fomenko atom 𝐀^**.




 §.§ Elliptic billiard with half-turn around the center


In this case, with φ is the half-turn about the center of the ellipse.
Such billiards are a case of the billiards with slipping, see <cit.>.

Examples of its trajectories are depicted in Figure <ref>.


As in the previous cases, we will first derive the divisor conditions for periodicity.


In the billiard table bounded by the ellipse  given by (<ref>), consider the magic billiard with half-turn about the center.
Consider a trajectory of such billiard with the caustic _β from the confocal family (<ref>).
Such a trajectory
is n-periodic if and only if nQ_0∼ nQ_∞.
Here Q_0 and Q_∞ are as in Theorem <ref>.


We follow the notation of Theorem <ref> and its proof, and, as there, we can derive the divisor condition (<ref>).
	
	In addition to that condition, we need that the number of times the particle hit the boundary, the number of times it crossed the long axis, and the number of times it crossed the short axis, are all odd or all even.
	This is equivalent to m and n being both odd or both even.
	
	Thus, the periodicity condition for even n is equivalent to n(Q_0-Q_c_1)∼0, and for odd n to n(Q_0-Q_c_1)+(Q_c_2-Q_a)∼0.
	Using 2Q_∞∼2Q_a∼2Q_b∼2Q_β and 3Q_∞∼ Q_a+Q_b+Q_β, we get that the periodicity condition for any n is nQ_0∼ nQ_∞.


From divisor conditions, one can derive the analytic conditions of Cayley's type and the polynomial conditions, similarly as in <cit.>.
We present them in the following two corollaries.


	In the billiard table bounded by the ellipse  given by (<ref>), consider the magic billiard with the half-turn about the center.
	Consider a trajectory of such billiard with the caustic _β from the confocal family (<ref>).
	Such a trajectory
	is n-periodic if and only if:
	
		
  * n is even and (<ref>) is satisfied; or
		
  * n is odd and

    |	[         B_2         B_3           …   B_(n+1)/2;         B_3         B_4           … B_(n+1)/2+1;           …;   B_(n+1)/2 B_(n+1)/2+1           …       B_n-1 ]|=0.

	 
The coefficients B_2, B_3, …are as in the statement of Corollary <ref>.



	The trajectories of the magic billiard with flipping over the long axis with caustic _β are n-periodic if and only if there exists a pair of real polynomials p_d_1, q_d_2 of degrees d_1, d_2 respectively, and satisfying the following:
	
		
  (a) if n=2m is even,  then d_1=m, d_2=m-2, and (<ref>);
		
  (b) if n=2m+1 is odd, then d_1=m, d_2=m-1, and
		
    sp_m^2(s)
    		-
    		(s-1/a)(s-1/b)(s-1/β)q_m-1^2(s)=1.

		


Finally, we give the topological description of this system.


	The rough Liouville equivalence class of an isoenergy manifold in the phase space of the magic billiard with flipping through the center is given by the Fomenko graph in Figure <ref>.



According to Remark <ref>, we define the configuration space as the billiard table where the points of  which are diametrically symmetric to each other are identified.
Such configuration space is homeomorphic to the projective plane.
	
In a fixed isoenergy manifold, any level set corresponding to a any given ellipse as caustic consists of two Liouville tori: one torus contains the trajectories that are winding in the clockwise direction about the caustic, the other torus contains the counterclockwise trajectories.

The level set corresponding to hyperbola as caustic also consists of two Liouville tori: one torus where all segments of the trajectories point downwards, and the other where they point upwards.
	
The level set corresponding to the caustic =_0  corresponds to a pair of Fomenko atoms 𝐀: each of them contains a single closed trajectory along the boundary ellipse, winding in one of the two directions.
	
The level set with the caustic _a corresponds to the motion the short axis of the ellipse .
It consists of two closed trajectories – the upwards and the downwards one.
Thus we have two Fomenko atoms of type 𝐀 there.
	
The level set with the caustic _b consists of the trajectories that contain the foci of the ellipse .
Two of those trajectories are closed: one corresponds to motion to the right along the long axis, the other to the motion to the left.
	
For any other trajectory on that level set, each segment will pass through the same focus, and each segment points in the same direction: upwards or downwards.
	Thus, there are four separatrices on that level set, so it is represented by the Fomenko atom 𝐂_2.



For a more detailed proof of Theorem <ref>, together with the calculation of the numerical invariants, see <cit.>.





§ MAGIC BILLIARDS IN ELLIPTIC ANNULUS


In this section, we discuss magic billiards in the annulus between two confocal ellipses _1 and _2, see Figure <ref>.
	
In order to preserve integrability and caustics, we will assume that the mapping φ is defined as follows:

	
  * the restriction of φ to _1 is the reflection with respect to one of the axes or the central symmetry;
	
  * the restriction to _2 is the identity.

The mapping φ^* will be defined on the velocity vectors at the points of _1 in the same way as explained in Section <ref>.
On _2, mapping φ^* is the ordinary billiard reflection.


If magic reflection is introduced on the inner boundary, then the dynamics does not any more depend continuously on the initial conditions.
Namely, consider motion parallel to a tangent line to _2, close to the point of tangency.
If the particle does not cross _2, the motion is continued straight along that line.
If the particle reaches _2, it will be, according to the mapping φ magically transported to another point of _2 and continue motion from there.
Thus, the continuity close to tangency to _2 is lost.



Two trajectories when φ on the outer boundary is flipping over the long axis are shown in Figure <ref>.




The magic billiard in the annulus between two confocal ellipses with flipping over the long axis on the outer boundary is roughly Liouville equivalent to the magic billiard within an ellipse with flipping over the short axis.


In a fixed isoenergy manifold, any level set corresponding to the trajectories having a fixed non-degenerate conic as caustic consists of a single Liouville torus.
	
The level set corresponding to the caustic =_0  corresponds to the Fomenko atom 𝐀, similarly as in the proof of Theorem <ref>.
	
The level set with the caustic _a corresponds to the motion along two segments of the short axis which are within the annulus.
Each time when the particle hits the outer boundary, it is magically flipped to the other segment, thus this level set consists of a single closed trajectory, so we have the Fomenko atom of type 𝐀 there.
	
The level set with the caustic _b consists of the trajectories that contain the foci of the ellipse .
Two of those trajectories is closed: each trajectory corresponds to the motion along one of the segments of the long axis which are within the annulus.
For any other trajectory on that level set, the extensions of the segments will pass alternately through the two foci.
In one class of those trajectories, the segments that point from the outer boundary to the inner one contain the left focus, and their limit as time goes to +∞ is the left segment on the long axis, while the time limit to -∞ is the right segment.
In the other class, everything is opposite.
Thus, there are two separatrices on that level set, so it is represented by the Fomenko atom 𝐀^**.



	Two trajectories when φ on the outer boundary is flipping over the short axis are shown in Figure <ref>.




	The magic billiard in the annulus between two confocal ellipses with flipping over the short axis on the outer boundary is roughly Liouville equivalent to the magic billiard within an ellipse with flipping over the long axis.


	In a fixed isoenergy manifold, any level set corresponding to the trajectories having an ellipse as caustic consists of a single Liouville torus.
	On the other hand, if the caustic is hyperbola, there are two tori: each corresponding to one connected component within the annulus between the two branches of the hyperbola.
	
	The level sets corresponding to the caustic =_0  corresponds to the Fomenko atom 𝐀.

	The level set with the caustic _a corresponds to the motion along two segments of the short axis which are within the annulus.
	Each segment is covered by one closed trajectory on that level set, thus both of them corresponds to the Fomenko atom 𝐀.
	
	The level set with the caustic _b consists of the trajectories that contain the foci of the ellipse .
	One of those trajectories is closed and it is placed on the long axis, traversing the two segments.
	Any other trajectory on that level set is placed on one side of the long axis, thus there are two separatrices.
We conclude that this level set corresponds to the the Fomenko atom 𝐁.



	Two trajectories when φ on the outer boundary is slipping by half-ellipse are shown in Figure <ref>.




The rough Liouville equivalence class of an isoenergy manifold in the phase space of the magic billiard in an elliptic annulus with slipping by half-ellipse along the outer boundary is given by the Fomenko graph in Figure <ref>.



	In a fixed isoenergy manifold, any level set corresponding to the trajectories having a fixed ellipse as caustic consists of two Liouville tori: each torus contains the trajectories the winding in one direction around the annulus. 
	On the other hand, if the caustic is hyperbola, there is only one torus.
	
	The level set corresponding to the caustic =_0 contains only two closed trajectories: each is winding in one direction along the boundary.
	Thus, we have two Fomenko atoms of type 𝐀 there.
	
	The level set with the caustic _a corresponds to the motion along two segments of the short axis which are within the annulus.
	There is only one trajectory on that level set, so it corresponds to the Fomenko atom 𝐀.
	
	The level set with the caustic _b consists of the trajectories that contain the foci of the ellipse .
	One of those trajectories is closed and it is placed on the long axis, traversing the two segments.
There are two separatrices on that level set: one separatrix contains the trajectories where the motion is above the long axis on the left and below the axis on the right as t→+∞, and vice versa for t→-∞.
The opposite holds for the other separatrix.
We can conclude that the corresponding Fomenko atom is 𝐁.



The example of billiard within elliptic annulus with slipping along the outer ellipse was analyzed in <cit.>.
There, the configuration space was defined differently: by gluing two identical annuli along inner and the outer boundaries, so the resulting Fomenko graph there is different.





§ CONCLUSIONS AND DISCUSSION


We note that the magic billiards, as introduced in Section <ref> represents a very broad class, where various subclasses may be of interest for exciting future research.

In particular, integrable cases for magic billiards within ellipse are not exhausted by the list considered in this paper. It would be interesting to explore a more general class of magic billiards (,φ,φ^*), including those which preserve caustic. One natural generalization includes studying domains with piece-wise smooth boundaries. 

We note that billiard ordered games introduced in <cit.> and studied further in <cit.> can also be generalised using magic reflections.





§ ACKNOWLEDGEMENTS
 
This paper is devoted to Academician Anatoly Timofeevich Fomenko on the occasion of his 80-th anniversary.  The authors have been glad to learn a lot from personal contacts with Anatoly Timofeevich and his distinguished students, from their lectures, books, and papers. The authors wish Anatoly Timofeevich many happy returns and further success in his scientific work and the work of his scientific school. 


This research is supported by the Discovery Project No. DP190101838
Billiards within confocal quadrics and beyond from the Australian Research Council, 
by  Mathematical Institute of the Serbian Academy of Sciences and Arts, the Science Fund of Serbia grant Integrability and Extremal Problems in Mechanics, Geometry and
Combinatorics, MEGIC, Grant No. 7744592 and the Ministry for Education, Science, and Technological Development of Serbia and the Simons Foundation grant no. 854861.

M. R. is grateful to Professors John Roberts and Wolfgang Schief and the School of Mathematics and Statistics of the UNSW Sydney for their hospitality and support during final stages of work on this paper.

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