Non-integrability of the planar elliptic restricted three-body problem

We present the non-integrability proof for the planar elliptic restricted three-body problem. Two versions of this problem are considered: the classical one when only gravitational interactions are taken into account, and the photo-gravitational version where radiation pressure from the primaries is also included. Our result is valid for nonzero eccentricity and arbitrary mass ratio of the primaries. In the proof, we apply the differential Galois approach to study the integrability.


Introduction
One of the fundamental model systems of celestial mechanics which has a plenty of applications is the restricted three-body problem. In this model, we assume that a point with a negligible mass moves in the gravity field of two other point masses called the primaries with masses m 1 and m 2 . They move in elliptic Keplerian orbits around their common mass centre. In the special case when ellipses become circles, we speak about the circular restricted three-body problem. As examples of the restricted three-body problem, we mention the Sun-Jupiter-asteroid or Sun-planet-object systems. In the case when an infinitesimal mass particle moves in the same plane as the orbits of the primaries, the problem is called planar elliptic restricted three-body problem, otherwise is called spatial one. We introduce the mass parameter μ = m 2 m 1 +m 2 , m 2 ≤ m 1 , μ ∈ (0, 1/2] and we assume that m 1 + m 2 = 1. Then the primaries have the following masses: the heavier one m 1 = 1 − μ and the lighter one m 2 = μ, respectively. The circular restricted three-body problem with μ = 1 2 is called the Copenhagen Problem after the series of papers published by Strömgren and colleagues in the Copenhagen Observatory Publication, see Szebehely and Nacozy (1967), Danby (1967) and references therein.
The problem of the integrability of the circular problem was investigated by many authors using different techniques starting from the famous memoir of Poincaré (1890) in which he proved the non-existence of an additional first integral which is analytic in coordinates, momenta and parameter μ for small μ. The dynamical reason of the non-integrability is the existence of periodic solutions with nonzero characteristic exponents, which originate from the breaking of periodic invariant tori of the integrable approximations. Poincaré improved and extended his result in Poincaré (1892, Chap. 5) where he started from the restricted planar circular problem, then passed to the unrestricted planar problem, and finally to the unrestricted spatial problem. Ideas and methods presented in these works were interpreted, extended and generalized by many authors. Using the Melnikov method Xia (1992) showed that for small enough μ, there exist transversal homoclinic orbits and this implies the nonexistence of any real analytic integral for all but possibly finite number of values of μ. This result was generalized in Guardia et al. (2016) where the occurrence of transverse intersection between the stable and unstable manifolds of the infinity (the notion introduced by authors) for any μ ∈ (0, 1) was shown. Recently, the non-integrability of this problem was also proved by means of differential Galois approach in Yagasaki (2021).
We know only few papers devoted to study the same question for the elliptic problem. The Arnold diffusion in the circular problem is prevented by KAM tori, but it is possible in the elliptic case, see Féjoz et al. (2016). For example, Xia (1993) proved the presence of the Arnold diffusion. His proof is based on the fact that the transversal homoclinic orbits in the circular restricted three-body problem exist. Capiński et al. (2016) used another approach based on shadowing of pseudo-orbits generated by two dynamics: an 'outer dynamics', given by homoclinic trajectories to a normally hyperbolic invariant manifold, and an 'inner dynamics', given by the restriction to that manifold. Other diffusive orbits in elliptic restricted three-body problem are described in Bolotin (2006). Along these orbits, the angular momentum changes in a bounded interval while trajectories come close to collisions. In Delshams et al. (2019), authors proved the existence of orbits whose angular momentum performs arbitrary excursions in a large region leading to global instability. In Guardia et al. (2017), Delshams et al. (2019), the existence of oscillatory orbits for the elliptic case and in Llibre and Simó (1980); Guardia et al. (2016) for the circular case, was proved. A trajectory is called oscillatory if it leaves every bounded region but returns infinitely often to some fixed bounded region.
The phenomena mentioned above show that the dynamics of the elliptic restricted threebody system is complex and in fact it is not integrable. All these results have been shown with applications of highly advanced analytical methods.
Our aim is to give a proof of the non-integrability of the planar elliptic restricted three-body problem. We consider two versions of this problem. In the classical version of the problem, only the gravitational interactions are taken into account, while in the photo-gravitational problem also the radiation pressure forces are included. We use methods which till now were not applied to study the integrability of the elliptic restricted problem.

Equations of motion
We assume that the considered three masses move in a plane. The primaries with masses m 1 and m 2 move in Keplerian elliptic orbits around their common mass centre, which is taken as the origin of the inertial barycentric and the rotating reference frames. The x-axis Fig. 1 Geometry of the restricted three-body problem in the rotating and pulsating frame of the rotating frame is directed from the more to the less massive primary. The coordinates are rescaled by the distance between the primaries. In this rotating and pulsating frame, the primaries are located at P 1 = (−μ, 0) and P 2 = (1 − μ, 0), respectively, see Fig. 1.
The motion of the third body with an infinitesimal mass is described by the following equations where f (e, ν) = 1 1 + e cos ν , and the effective potential is The independent variable in these equations is the true anomaly ν. For details, see, Szebehely (1967, Sect. 10.3), Gawlik et al. (2009). We fix the units in such a way that m 1 + m 2 = 1, the gravitational constant G = 1, the semi-major axis of the relative orbit of the primaries a = 1.
As it is well known, the system has five equilibria, called the Lagrange points. In our considerations, we will use only triangular libration point L 4 = 1 2 − μ, √ 3 2 . In the inertial frame, this is just an elliptic orbit and its eccentricity and the semi-major axis are the same as the relative orbit of the primaries.
The photo-gravitational elliptic restricted three-body problem is a generalization of the above described system when an infinitesimal mass is affected not only by gravitation but also by radiation pressure from the primaries. We assume that both the primaries are stars radiating a constant amount of the light and the infinitesimal mass is a spherical particle with a uniform albedo. Since the radiation pressure force F r changes with the distance according to the same inverse square law as the gravity force F g but with opposite sign, the total force exerted by a primary is F = F g − F r = F g (1 − β), where β = F r /F g and the effect of the radiation pressure is the same as that of reducing the stellar mass by a term β, see e.g. Chernikov (1970), Todoran and Roman (1993). The tidal and rotational distortions of the radiating primaries are here neglected. If we introduce σ 3 i = 1 − β i , i = 1, 2 for the respective primaries, then the equations of motion of the infinitesimal mass particle are the following d 2 q 1 dν 2 − 2 (2) The constants are σ i ∈ [0, 1], where σ i = 1 corresponds the previously described case without radiation, while σ i = 0 corresponds to the situation when radiation pressure force balances the gravitational force. Also the photo-gravitational problems with only one radiating body are considered as it was in the first article concerning the photo-gravitation restricted threebody problem of Radzievskii (1950), whereas the primaries were considered: the Sun and a planet, and a dust particle as an infinitesimal mass. For a discussion of different versions of this problem see Kunitsyn and Polyakhova (1995). If σ 1 + σ 2 ≥ 1, then there exist triangular libration points, and, in this case L 4 = (q 0 1 , q 0 2 ), where Thus, the distances between the primaries and the libration point are so, assumption σ 1 + σ 2 ≥ 1 is just the triangle inequality because the distance between the primaries is 1. Let us notice that the equations of motion (2) as well as its particular case (1) are Hamiltonian. In fact, they are equivalent to the canonical equations generated by the following Hamiltonian function Let us notice that Hamilton equations governed by (4) have an additional first integral L = p 1 q 2 − p 2 q 1 in the following cases: when either μ = 0 or μ = 1, i.e. one of the primaries vanish, or when σ 1 = σ 2 = 0, i.e. when radiation pressure forces of both the primaries balance their gravitational interactions. The Hamiltonian system generated by (4) is not autonomous (ν is the independent variable); however, it is 2π periodic.

Non-integrability theorem
Our aim is to study the integrability of the Hamiltonian system given by (4). At first, we recall some basic facts about the integrability. An autonomous Hamiltonian system with n degrees of freedom is given by its Hamiltonian function H (q, p), where q = (q 1 , . . . , q n ) and p = ( p 1 , . . . , p n ) are the canonical coordinates and momenta, respectively. A function F(q, p) is a first integral of this system if it is constant along all solutions of the Hamilton equations denotes the Poisson bracket. We say that the system is integrable in the Liouville sense, or that it is completely integrable if it admits n functionally independent first integrals F 1 , . . . , F n which pairwise commute, that is, The integrability of the system implies that it is solvable by quadratures. For a detailed exposition, see, Arnold (1989), Arnold et al. (1988). If the Hamiltonian of the system depends explicitly on time H = H (q, p, t), then we can introduce an additional canonical pair of variables (q n+1 , p n+1 ) and Hamiltonian function K = H (q, p, p n+1 ) − q n+1 . Then, the last pair of equations of motion reads, Hence, p n+1 is the time. We say that the non-autonomous Hamiltonian system given by Hamiltonian H (q, p, t) is completely integrable, if the system with (n + 1) degrees of freedom given by Hamiltonian K = H (q, p, p n+1 ) − q n+1 is completely integrable. For a detailed explanation, we refer here to the paper of Kozlov (1983). In our investigation of the integrability of the elliptic restricted problem, we apply a method which is based on the study of the variational equations of the considered system along a particular solution. The variational equations are Hamiltonian and if the original system is integrable, then the variational system is also integrable. The variational equations are linear, and their integrability puts strong restrictions on the properties of the monodromy and the differential Galois groups which are attached to them. These arguments are just a starting point for the Ziglin and the Morales-Ramis theories, see Morales Ruiz (1999). The fundamental result of this approach is the Morales-Ramis theorem.

Theorem 1 Assume that a complex Hamiltonian system with n degrees of freedom is integrable with complex meromorphic first integrals in the Liouville sense in a neighbourhood of a phase curve . Then, the identity component of the differential Galois group of the variational equations along is Abelian.
For details, see the paper of Morales Ruiz (1999). Although this theorem is based on involved mathematical theory, it appears that it can be effectively applied to study the integrability of a wide variety of systems from dynamical astronomy, physics and other branches of applied sciences. It is enough to mention here the problem of the integrability of the three-body problem which was solved thanks to the application of this theory, see articles Boucher and Weil (2003), Maciejewski and Przybylska (2011), and also Tsygvintsev (2000), Tsygvintsev (2001) where the monodromy group that is a subgroup of the differential Galois group was used. Moreover, it appears that for application of this theorem, there are algorithms accessible for a wide audience. A practical introduction to the subject and numerous applications of the above theorem can be found in Morales-Ruiz and Ramis (2010).
In the above theorem, it is assumed that the Hamiltonian as well as the first integrals which guarantee the integrability are complex meromorphic functions. In the considered case, Hamiltonian (4) is not a meromorphic function because it contains terms with radicals. However, as it was explained in Combot (2013), Maciejewski and Przybylska (2016), still one can apply the Morales-Ramis theory for systems with algebraic potentials.
We formulate our main result in the following two theorems. The first one concerns the classical restricted problem.
The second theorem concerns the photo-gravitational problem.
Notice that in the above theorem, we assume the strict inequality (9), although the triangular libration point L 4 exists when σ 1 + σ 2 = 1. We did not exclude this case accidentally. The reason is that, as we demonstrate it later, for this particular case the necessary conditions are fulfilled. An investigation of the integrability in this case needs stronger tools. Here, we remark only that in the case σ 1 + σ 2 = 1, the triangular point is, in fact, a collinear one. The above theorems also prove the non-integrability of the spatial version of the restricted problem as the former is an invariant subsystem of the latter.
In the proof of this theorem, we will use the Morales-Ramis Theorem 1. As a particular solution, we take the triangular libration point L 4 . This is why, we need the assumption (9).

Variational equations
We consider our problem in the extended phase space with coordinates (q 1 , q 2 , q 3 ) and the conjugated momenta ( p 1 , p 2 , p 3 ). We set z = (q, p, q 3 , p 3 ), where q = (q 1 , q 2 ) and p = ( p 1 , p 2 ). In this space, equations of motion of the system are generated by Hamiltonian They have the following forṁ These equations admit the particular solution The variational equations along this solution have the form where and As it is clearly visible, the system (14) splits into two subsystems. The first one, called the normal variational equation, corresponds to the first 4 × 4 block, the second one, called the tangential equation, is given by the 2 × 2 block. In our study of the integrability, it is enough to investigate the normal variational equation, that is where We perform a further simplification of this equation. To this end, we first make a transformation with constant coefficients. Namely, we put x = C z, where z = [z 1 , z 2 , z 3 , z 4 ] T denotes the original variables, and where ϕ is a parameter which we will fix later. As a result, we obtain where V = R(ϕ)V R(ϕ) −1 . Because the Hessian matrix V is real and symmetric, one can find a rotation matrix R(ϕ) which transforms it to the diagonal form. Thus, we can assume that the matrix I − V is diagonal. Direct calculations give where For the classical purely gravitational problem, the parameter δ simplifies to Notice that the case δ = 0 is impossible in the purely gravitational problem, and in the photo-gravitational problem, it occurs when μ = 1/2 and σ 2 1 + σ 2 2 = 1. In the lemmas below, the value δ = 1 is excluded. Again, this case is impossible in the purely gravitational problem. It occurs only when g = 0. As by assumption μ(1 − μ) = 0, according to (22) we have Thus, one factor in the above formula vanishes. However, as σ 1 , σ 2 ∈ (0, 1], there is only one possibility, namely σ 1 + σ 2 = 1. In this case, L 4 coincides with a collinear libration point L 1 . The above simplification of the variational equation for the triangular libration points in the elliptic restricted three-body problem is well known, see for example Szebehely (1967, Sect. 10.3) or Grebenikov (1964). For the triangular libration points of the photo-gravitational elliptic restricted three-body problem, it can be found in Markellos et al. (1992).
If we introduce the entries of the Hessian matrix then the parameter ϕ determining the orthogonal transformation R(ϕ) diagonalizing V can be obtained from the vanishing of off-diagonal elements of the transformed matrix V = R(ϕ)V R(ϕ) −1 which gives The transformed normal variational equation depends only on two parameters: e ∈ (0, 1) and δ ∈ [0, ∞). However, planning an application of the Morales-Ramis theory we need to our disposal tools which allow determining the monodromy and the differential Galois group of this parameter-dependent equations. Unfortunately, for systems of four equations, there are no sufficiently strong results. This is why we will continue a simplification of the equation.
In the sixties of the previous century, the analysis of the stability of libration points was a popular subject, see Danby (1964), Grebenikov (1964), Bennett (1965), Alfriend and Rand (1969), Tschauner (1971), Meire (1981). Just for the needs of these investigations, Tschauner (1971) found a time dependent transformation which splits the systems of four coupled equations into two uncoupled second-order equations. Later Meire (1981), Matas (1982), Matas (1973) continued the study of this problem in more details. All of these results concerns only the classical elliptic restricted problem. At first, we assume that δ = 0, and let us introduce the following quantities Q(e, δ) = 9δ 4 − 8δ 2 + e 4 + 2δ 2 e 2 , = Q(e, δ), and matrices where −2e sin(ν)(δ − e cos(ν)) −3δ 2 + e 2 cos(2ν) −3δ 2 + e 2 cos(2ν) −2e sin(ν)(δ + e cos(ν)) + J. Now, the transformation given by is non-singular if Q(e, δ) = 0. In fact, Using this transformation, we obtain where The above transformation is just a simple generalization of the transformation given by Tschauner (1971), see also Markellos et al. (1993). The values of the parameters that lie on the curve Q(e, δ) = 0, see solid line curve in Fig. 2, are excluded in the above considerations. However, we cannot exclude them from the study of the integrability. We did not find any investigation of this particular case. The question if the variational equations split in these cases was unknown. This is why we decided to investigate it with the help of differential algebra tools, see Compoint and Weil (2004). Omitting the details, the result is the following.
For this particular case, matrix V is diagonal and the matrix of normal variational equations given in (20) simplifies to Now, the change of variables y = T (ν)x with the constant matrix transforms the system (20) into (30) where the block diagonal matrix B(ν) given by (31) has the following blocks

Non-integrability
In this section, we outline the proof of Theorem 3. By Theorem 1, if the system is integrable, then the identity component of the differential Galois group of variational equations (14) is commutative. We also say that the differential Galois group is virtually commutative. Anyway, our plan is to show that this group is not virtually commutative. The important fact is that if the differential Galois group of a system is virtually commutative, then the differential Galois group of each of its subsystem is also virtually commutative. From this fact, it follows that it is enough to investigate the normal variational equation (18) which after simple change of variable has a particularly simple form (20).
As we showed in the previous section, a further simplification is possible. Namely, after transformation (28), the system has a block diagonal form (30). Hence, it is enough to investigate a subsystem corresponding to one of these blocks. We choose Elements of matrix B + (ν) are periodic functions of ν. Algorithms which we are going to apply assume that the coefficients of the considered system are rational functions of the independent variable. This is why we introduce the new independent variable z = e iν . After this transformation, we obtain As matrix C + (z) is rational as required. Now we can claim the following.
The proof of this lemma is given in "Appendix A". In this way, we proved our Theorem 3 except the cases when Q(e, δ) = 0.
Let us assume now that Q(e, δ) = 0. Then, as we have shown in the previous section, the variational equations can be transformed to the block-triangular form (34). Hence, for further considerations, we take the subsystem corresponding to the block B 0 (ν), see (27), and we make the same change of independent variable as in the previous case. In effect, we obtain the following system Elements of the matrix C 0 (z) are rational functions of z. The final step of our reasoning is the following lemma.
We prove this lemma in "Appendix A". Finally, we consider the particular case excluded in the previous lemmas when δ = 0. Then, the variational equations can be also transformed to the block-triangular form (34) with blocks B + , B − given in (37). In the next step, we rationalize the subsystem corresponding to B + defining matrix of linear system with rational coefficients C + (z) according to (39). Analysis of the differential Galois group of these linear differential equations gives the last lemma.
The proof of this lemma is also contained in "Appendix A". In effect, we have shown that for all assumed values of the parameters the differential Galois group of a subsystem of the variational equations is not virtually commutative, so the differential Galois group of the variational equations is not virtually commutative. Hence, by the Morales-Ramis Theorem 1, the system is not integrable.

Conclusions
In this paper, we have presented the non-integrability proof of the elliptic restricted threebody problem for arbitrary nonzero eccentricity e ∈ (0, 1) and arbitrary ratio of masses of the primaries μ ∈ 0, 1 2 . Analysis was made for the classical case when only gravitational influence of the primaries on the test particle is considered and also for the photo-gravitational version when also the radiation pressure from the primaries is included. This generalization introduces two additional parameters which measure the strength of the radiation forces from two primaries. We prove the non-integrability of this problem for all allowable values of parameters such that the triangular libration point exists.
The proof uses properties of the differential Galois group of four dimensional variational equations along a particular solution determined by the triangular libration point L 4 . When checking the differential Galois group of variational equations, their factorization into two 2D subsystems turned out to be very useful.
The fact that we succeeded to prove the non-integrability of the photo-gravitational problem for a whole domain of four parameters is in some sense exceptional. Typically, for certain values of parameters, the necessary conditions are fulfilled. We showed that the variational equations for both, classical and photo-gravitational problem, depend only on two parameters e and δ, and this is why the complete integrability analysis was performed for both versions simultaneously.
We show that the method used for proving the non-integrability when applied to study the considered system reduces to a quite simple algorithm which can be applied for other problems.
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Appendix A: Proofs
Here, we present the proofs of Lemmas 4, 5 and 6 . All of them are just direct applications of Lemma 7 which we formulated in "Appendix B". This Lemma, follows from the Kovacic algorithm (Kovacic 1986). It is described in many places, see, for example Morales-Ruiz (1989), Duval and Loday-Richaud (1989). So here we only recall that this algorithm gives definitive answer when a second-order differential equation with rational coefficient is solvable in terms of Liouvillian functions. If it gives the negative answer, then the differential Galois group of this equation is not virtually commutative. This is why it is used frequently in cases when for the integrability studies of a system the Morales-Ramis Theorem 1 is applied. Before we pass to the proofs, we recall that a system of two linear equations y 1 = a 11 (z)y 1 + a 12 (z)y 2 , y 2 = a 21 (z)y 1 + a 22 (z)y 2 , (A.2) can be transformed into a second-order differential equation of the form (A.1). This transformation is not unique. In general, we take a linear combination w = α 1 y 1 + α 2 y 2 with constant coefficients α 1 and α 1 . Then, we calculate w , and using it together with (A.2) we can express w in terms of w and w . This procedure works for a 'generic' choice of (α 1 , α 2 ) and different choices of these constants give different coefficients p(z) and q(z).

A.1. Proof of Lemmas 4 and 5
First, we transform system (39) to the second-order equation. We put w = y 1 + iy 2 . Using the above described procedure, we obtain an equation of the form (A.1) with In order to apply Lemma 7, we transform this equation to the reduced form At first, we assume that they are pairwise different. That is, that the discriminant disc(d(z)) of the polynomial does not vanish. Direct calculations give disc(d(z)) = −48δ 2 e 6 e 2 − 1 9δ 4 − 12δ 2 + e 4 + 6δ 2 e 2 2 .
As e, δ ∈ (0, 1), this discriminant vanishes if the polynomial η(e, δ) = e 2 + 3δ 2 2 − 12δ 2 (A.10) vanishes. Hence, at first we assume that η(e, δ) = 0. Even under this assumption, it can happen that for certain values of parameters, the numerator and the denominator of r (z) are not relatively prime. Then, the order of singular points can depend on parameters. To check this, we have to calculate the resultant of the numerator and denominator of r (z). If they have a common factor, then the resultant vanishes. It appears that for the assumed values of the parameters, it vanishes only along the curve η(e, δ) = 0. A justification of this fact is direct, but rather long, so we omit it. The function r (z) has the following expansion at singular points Moreover, its expansion at infinity reads Hence, all singularities of equation (A.6) are regular. The differences of exponents i at the respective singularities are 0 = 1 = 2 = ∞ = 2, 3 = 4 = 1.
Thus, it is possible that the logarithmic terms are present in local solutions near a singularity, see explanations above equation (B.7) in "Appendix B". In fact, such a term can be found for a solution near z = 0. One local solution corresponding to the exponent ρ + = 3/2 has the form One can find easily that so, according to formula (B.8), the coefficient g 2 multiplying the logarithmic term in local solution is As δ ∈ (0, ∞) \ {1}, g 2 = 0 and the logarithmic term is always present. Now, we show that equation (A.4) does not admit a nonzero hyperexponential solution. For the definition of this notion and its calculations, see "Appendix B" starting from the paragraph above Lemma 7. According to equation (B.9), we have to select all possible choices of exponents ε = (ε 0 , . . . , ε 4 , ε ∞ ) such that is a non-negative integer. Because z 3 and z 4 are poles of the first order, then ε 3 = ε 4 = 1, see "Appendix B". For the remaining singular points we have In effect there is only one choice which gives n = 1, and the corresponding function ω(z) defined in (B.9) is Inserting the polynomial P(z) = z + p 0 into equation (B.11), we obtain a system of three algebraic equations for ( p 0 , e, δ) which has a solution only when either e = 0 or δ = ±1. These values of parameters are excluded, thus there is no polynomial P(z) of degree one which satisfies equation (B.11). In effect, Eq. (A.4) does not admit a hyperexponential solution, and its differential Galois group is not virtually commutative. The above conclusion is valid under the assumptions that η(e, δ) = 0, see (A.10). Polynomial η(e, δ) factorizes as η(e, δ) = η + (e, δ)η − (e, δ), η ± (e, δ) = e 2 + 3δ 2 ± 2 √ 3δ. (A.14) The curve η + (e, δ) = 0 does not have a real component, so it is enough to investigate the curve η − (e, δ) = 0, see dashed curve in Fig. 2. This curve admits the following rational parametrization e = 2s 1 + s 2 , δ = 2 √ 3s 2 3(1 + s 2 ) . and the reduced form (A.4) of this equation has coefficient . (A.21) The condition for coalescence of singularities is e(e 2 − 1) = 0, thus by assumptions we always have four singularities: one pole of the second-order z 0 = 0, two poles of the firstorder z 1,2 = −1± It means that g 2 = 0 and the logarithmic term in the local solutions is always present. Now we check whether equation (A.4) admits a nonzero hyperexponential solution. We have only one choice of ε = (ε 0 , ε 1 , ε 2 , ε ∞ ) such that n = ε ∞ − (ε 0 + ε 1 + ε 2 ), is a non-negative integer. For poles of the first-order, ε 1 = ε 2 = 1 and the remaining exponents belong to the set This gives only one choice ε = − 1 2 , 1, 1, 3 2 , with n = 0 and Without loss of generality, we can take P(z) = 1 and substitution into Eq. (B.11) gives contradiction 1 = 0. Thus, also in this case, a subset of normal variational equations does not have a hyperexponential solution and its differential Galois group is full SL(2, C) which gives the statement of Lemma 6.
subgroup of SL(2, C). The presence of a logarithmic singularity allows to formulate a simple and effective criterion. Before its formulation, we recall that the function w(z) is called hyperexponential if w (z)/w(z) is a rational function. In other words, it is of the form w(z) = exp[ a(z)dz], where a(z) is a rational function.
Lemma 7 Assume that the differential equation (B.1) has a logarithmic singularity, and it has no nonzero hyperexponential solution, then its differential Galois group is SL(2, C).
Let us describe shortly the algorithm which allows to find the hyperexponential solution of Fuchsian equation (B.1), if it exists. In fact, it is the first case of the Kovacic algorithm (Kovacic 1986).
The hyperexponential solution that we look for has the following form where P(z) is a polynomial, z i are poles of r (z) of the first or the second order, and ε i is an exponent corresponding to this singularity. For each second order pole we have two choices for ε given by equation (B.6), and for each first order pole there is only one choice ε = 1. Polynomial P(z) is of degree n, where and ε ∞ is the exponent at infinity. Here, the sum is taken over all poles of r (z). Moreover, the polynomial P(z) has to be the solution of the following equation P + 2ω(z)P + (ω (z) + ω(z) 2 − r (z))P = 0. (B.11) In practice as a candidate for P(z) we take a polynomial of degree n with indefinite coefficients and insert it into the above equation. Then, the problem reduces to finding solutions of linear equations for these coefficients. For an equation with m singular points, we have 2 m+1 possible choices of elements (ε 1 , . . . , ε m , ε ∞ ), so we have at most such number of possibilities for choice of n and ω(z). For the proof of the above Lemma and its numerous applications, we refer the reader to Morales Ruiz (1999).