On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

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May 2015, 35(5): 1767-1800. doi: 10.3934/dcds.2015.35.1767

On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

Primitivo B. Acosta-Humánez ^1, , J. Tomás Lázaro ^2, , Juan J. Morales-Ruiz ^3, and Chara Pantazi ^2,

1.	Department of Mathematics, Universidad del Atlántico and Intelectual.Co, Barranquilla, Colombia
2.	Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Barcelona, Spain, Spain
3.	Department of Applied Mathematics, Technical University of Madrid, Madrid, Spain

Received September 2013 Revised September 2014 Published December 2014

Abstract
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We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached.

Keywords: Poincaré problem, rational first integral, Riccati equation, Differential Galois theory, integrating factor, Liouvillian solution., Liénard equation, Darboux theory of integrability.

Mathematics Subject Classification: Primary: 12H05; Secondary: 32S6.

Citation: Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz, Chara Pantazi. On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1767-1800. doi: 10.3934/dcds.2015.35.1767

References:

[1]	P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics, PhD. Thesis, Universitat Politecnica de Catalunya, 2009, arXiv:0906.3532.
[2]	P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics. The Integrability Analysis of the Schrödinger Equation by Means of Differential Galois Theory, VDM Verlag Dr. Müller, Berlin, 2010.
[3]	P. B. Acosta-Humánez, La teoría de Morales-Ramis y el algoritmo de Kovacic, Lecturas Matemáticas, 27 (2006), 21-56.
[4]	P. B. Acosta-Humánez, Nonautonomous Hamiltonian systems and Morales-Ramis theory I. The case $\ddot x=f(x,t)$, SIAM Journal on Applied Dynamical Systems, 8 (2009), 279-297. doi: 10.1137/080730329.
[5]	P. Acosta-Humánez and D. Blázquez-Sanz, Non-integrability of some Hamiltonians with rational potential, Discrete Continuous Dynam. Systems - B, 10 (2008), 265-293. doi: 10.3934/dcdsb.2008.10.265.
[6]	P. B. Acosta-Humánez, J. J. Morales-Ruiz and J.-A. Weil, Galoisian approach to integrability of Schrödinger equation, Report on Mathematical Physics, 67 (2011), 305-374. doi: 10.1016/S0034-4877(11)60019-0.
[7]	P. B. Acosta-Humánez and C. Pantazi, Darboux integrals for Schrödinger planar vector fields via Darboux transformations, SIGMA Symmetry Integrability Geom. Methods Appl., 8 (2012), Paper 043, 26 pp. doi: 10.3842/SIGMA.2012.043.
[8]	P. B. Acosta-Humánez, A. Reyes-Linero and J. Rodríguez-Contreras, Algebraic and qualitative remarks about the family $yy'=(\alpha x^{m+k-1}+\beta x^{m-k-1})y+\gamma x^{2m-2k-1}$, preprint 2014.
[9]	F. Baldassarri, On algebraic solutions of Lamé's differential equation, J. Diff. Equat., 41 (1981), 44-58. doi: 10.1016/0022-0396(81)90052-8.
[10]	D. Blázquez-Sanz, Differential Galois Theory and Lie-Vessiot Systems, VDM Verlag Dr. Müller, Berlin, 2008.
[11]	D. Blázquez-Sanz and J.-J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in Differential algebra, complex analysis and orthogonal polynomials, (eds P.B. Acosta-Humánez and F. Marcellán), Contemp. Math., Amer. Math. Soc., Providence, 509 (2010), 1-58. doi: 10.1090/conm/509.
[12]	D. Blázquez-Sanz and Ch. Pantazi, A note on the Darboux theory of integrability of non-autonomous polynomial differential systems, Nonlinearity, 25 (2012), 2615-2624. doi: 10.1088/0951-7715/25/9/2615.
[13]	M. Carnicer, The Poincaré problem in the nondicritical case, Ann. Math., 140 (1994), 289-294. doi: 10.2307/2118601.
[14]	G. Casale, Feuilletages singuliers de codimension un, groupoide de Galois et intègrales premieres, Ann. Inst. Fourier, 56 (2006), 735-779. doi: 10.5802/aif.2198.
[15]	D. Cerveau and A. Lins Neto, Holomorphic foliations in $\mathbbC \mathbbP (2)$ having an invariant algebraic curve, Ann. Inst. Fourier, 41 (1991), 883-903. doi: 10.5802/aif.1278.
[16]	E. S. Cheb-Terrab, Solutions for the general, confluent and biconfluent Heun equations and their connection with Abel equations, J. Phys. A, 37 (2004), 9923-9949. doi: 10.1088/0305-4470/37/42/007.
[17]	E. S. Cheb-Terrab and A. D. Roche, An Abel ordinary differential equation class generalizing known integrable classes, European J. Appl. Math., 14 (2003), 217-229. doi: 10.1017/S0956792503005114.
[18]	T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978.
[19]	C. Christopher, Invariant algebraic curves and conditions for a center, Proc. Roy.Soc. Edinburgh Sect. A, 124 (1994), 1209-1229. doi: 10.1017/S0308210500030213.
[20]	C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Darboux integrating factors: Inverse problems, J. Diff. Equat., 250 (2011), 1-25. doi: 10.1016/j.jde.2010.10.013.
[21]	C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Inverse problems for multiple invariant curves, Proc. Roy.Soc. Edinburgh Sect. A, 137 (2007), 1197-1226. doi: 10.1017/S0308210506000400.
[22]	C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Inverse problems for invariant algebraic curves: Explicit computations, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 287-302. doi: 10.1017/S0308210507001175.
[23]	C. Christopher, J. Llibre, Ch. Pantazi and X. Zhang, Darboux integrability and invariant algebraic curves for planar polynomial systems, J. Phys. A, 35 (2002), 2457-2476. doi: 10.1088/0305-4470/35/10/310.
[24]	C. Christopher, J. Llibre and J. V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math., 229 (2007), 63-117. doi: 10.2140/pjm.2007.229.63.
[25]	G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. math. 2ème série, 2 (1878), 60-96; 123-144; 151-200.
[26]	V. A. Dobrovol'skii, N. V. Lokot and J.-M. Strelcyn, Mikhail Nikolaevich Lagutinkskii (1871-1915): un mathématicien méconnu, Historia Math., 25 (1998), 245-264. doi: 10.1006/hmat.1998.2194.
[27]	A. Duval and M. Loday-Richaud, Kovacic's algorithm and its application to some families of special functions, Appl. Algebra Engrg. Comm. Comput., 3 (1992), 211-246. doi: 10.1007/BF01268661.
[28]	G. H. Halphen, Traité des Fonctions Elliptiques, Vol. I, II, Gauthier-Villars, Paris, 1888.
[29]	I. A. García, H. Giacomini and J. Giné, Generalized nonlinear superposition principles for polynomial planar vector fields, J. Lie Theory, 15 (2005), 89-104.
[30]	H. Giacomini , J. Giné and M. Grau, Integrability of planar polynomial differential systems through linear differential equations, Rocky Mountain J. Math., 36 (2006), 457-485. doi: 10.1216/rmjm/1181069462.
[31]	E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944.
[32]	J. P. Jouanolou, Equations de Pfaff Algébriques, Lectures Notes in Mathematics, 708, Springer-Verlag, New York/Berlin, 1979. doi: 10.1007/BFb0063393.
[33]	I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1957.
[34]	T. Kimura, On Riemann's equations which are solvable by quadratures, Funkcialaj Ekvacioj, 12 (1969), 269-281.
[35]	J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symb. Comput., 2 (1986), 3-43. doi: 10.1016/S0747-7171(86)80010-4.
[36]	E. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, Pure and Applied Mathematics, 54, New York - London, 1973.
[37]	A. Lins Neto, Construction of singular holomorphic vector fields and foliations in dimension two, J. Differential Geometry, 26 (1987), 1-31.
[38]	J. Llibre and Ch. Pantazi, Polynomial differential systems having a given Darbouxian first integral, Bull. Sci. Math, 128 (2004), 775-788. doi: 10.1016/j.bulsci.2004.04.001.
[39]	J. Llibre and Ch. Pantazi, Darboux theory of integrability for a class of nonautonomous vector fields, J. Math. Phys., 50 (2009), 102705, 19pp. doi: 10.1063/1.3205450.
[40]	J. Llibre, Ch. Pantazi and S. Walcher, Morphisms and inverse problems for Darboux integrating factors, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1291-1302. doi: 10.1017/S0308210511001430.
[41]	J. Llibre and C. Valls, Analytic integrability of quadratic-linear polynomial differential systems, Ergod. Th. & Dynam. Sys., 31 (2011), 245-258. doi: 10.1017/S0143385709000868.
[42]	J. Llibre and X. Zhang, Darboux theory of integrability in $\C^n$ taking into account the multiplicity, J. Diff. Equat., 246 (2009), 541-551. doi: 10.1016/j.jde.2008.07.020.
[43]	J. Llibre and X. Zhang, Darboux theory of integrability for polynomial vector fields in $\mathbb R^n$ taking into account the multiplicity at infinity, Bull. Sci. Math., 133 (2009), 765-778. doi: 10.1016/j.bulsci.2009.06.002.
[44]	B. Malgrange, Le groupoide de Galois d'un feuilletage, in Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., 38 (2001), 465-501.
[45]	B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, 23 (2002), 219-226. doi: 10.1142/S0252959902000213.
[46]	J. Martinet and J. P. Ramis, Theorie de Galois differentielle et resummation, in Computer Algebra and Differential Equations, (ed. E. Tournier), Comput. Math. Appl., Academic Press, London, (1990), 117-214.
[47]	J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems, Progress in Mathematics, 179, Birkhäuser, Basel, 1999. doi: 10.1007/978-3-0348-8718-2.
[48]	Ch. Pantazi, Inverse Problems of the Darboux Theory of Integrability for Planar Polynomial Differential Systems, PhD. thesis, Universitat Autonoma de Barcelona, 2004.
[49]	J. V. Pereira, Vector fields, invariant varieties and linear systems, Annales de l'institut Fourier, 51 (2001), 1385-1405. doi: 10.5802/aif.1858.
[50]	J. V. Pereira, On the Poincaré problem for foliations of general type, Math. Ann. , 323 (2002), 217-226. doi: 10.1007/s002080100277.
[51]	A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Second Edition, Chapman and Hall, Boca Raton, 2003.
[52]	E. G. C. Poole, Introduction to the Theory of Linear Differential Equations, Dover Publications, Inc., New York, 1960.
[53]	M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc., 279 (1983), 215-229. doi: 10.1090/S0002-9947-1983-0704611-X.
[54]	M. van der Put and M. Singer, Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer Verlag, Berlin, 2003. doi: 10.1007/978-3-642-55750-7.
[55]	M. F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc., 333 (1992), 673-688. doi: 10.1090/S0002-9947-1992-1062869-X.
[56]	M. F. Singer, Liouvillian solutions of $n$th order homogeneous linear differential equations, Amer. J. Math., {103} (1981), 661-682. doi: 10.2307/2374045.
[57]	F. Ulmer and J. A. Weil., Note on Kovacic's algorithm, J. Symb. Comp., 22 (1996), 179-200. doi: 10.1006/jsco.1996.0047.
[58]	R. Vidunas, Differential equations of order two with one singular point, J. Symb. Comp., 28 (1999), 495-520. doi: 10.1006/jsco.1999.0312.
[59]	J.-A. Weil, Constantes et Polynômes de Darboux en Algèbre Differentielle: Applications Aux Systèmes Différentiels Linéaires, PhD. Thesis, École Politechnique, 1995.
[60]	J. A. Weil, Recent algorithms for solving second-order differential equations, in Algorithms Seminar 2001-2002, (ed. F. Chyzak), INRIA, (2003), 43-46.
[61]	E. T. Whittaker and E. T. Watson, A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions: With an Account of the Principal Transcendental Functions, Cambridge Univ. Press, New York, 1962.
[62]	H. Zołądek, Polynomial Riccati equations with algebraic solutions, in Differential Galois theory (Bedlewo, 2001), Banach Center Publ., 58, Polish Acad. Sci., Warsaw, (2002), 219-231. doi: 10.4064/bc58-0-17.

show all references

References:

[1]	P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics, PhD. Thesis, Universitat Politecnica de Catalunya, 2009, arXiv:0906.3532.
[2]	P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics. The Integrability Analysis of the Schrödinger Equation by Means of Differential Galois Theory, VDM Verlag Dr. Müller, Berlin, 2010.
[3]	P. B. Acosta-Humánez, La teoría de Morales-Ramis y el algoritmo de Kovacic, Lecturas Matemáticas, 27 (2006), 21-56.
[4]	P. B. Acosta-Humánez, Nonautonomous Hamiltonian systems and Morales-Ramis theory I. The case $\ddot x=f(x,t)$, SIAM Journal on Applied Dynamical Systems, 8 (2009), 279-297. doi: 10.1137/080730329.
[5]	P. Acosta-Humánez and D. Blázquez-Sanz, Non-integrability of some Hamiltonians with rational potential, Discrete Continuous Dynam. Systems - B, 10 (2008), 265-293. doi: 10.3934/dcdsb.2008.10.265.
[6]	P. B. Acosta-Humánez, J. J. Morales-Ruiz and J.-A. Weil, Galoisian approach to integrability of Schrödinger equation, Report on Mathematical Physics, 67 (2011), 305-374. doi: 10.1016/S0034-4877(11)60019-0.
[7]	P. B. Acosta-Humánez and C. Pantazi, Darboux integrals for Schrödinger planar vector fields via Darboux transformations, SIGMA Symmetry Integrability Geom. Methods Appl., 8 (2012), Paper 043, 26 pp. doi: 10.3842/SIGMA.2012.043.
[8]	P. B. Acosta-Humánez, A. Reyes-Linero and J. Rodríguez-Contreras, Algebraic and qualitative remarks about the family $yy'=(\alpha x^{m+k-1}+\beta x^{m-k-1})y+\gamma x^{2m-2k-1}$, preprint 2014.
[9]	F. Baldassarri, On algebraic solutions of Lamé's differential equation, J. Diff. Equat., 41 (1981), 44-58. doi: 10.1016/0022-0396(81)90052-8.
[10]	D. Blázquez-Sanz, Differential Galois Theory and Lie-Vessiot Systems, VDM Verlag Dr. Müller, Berlin, 2008.
[11]	D. Blázquez-Sanz and J.-J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in Differential algebra, complex analysis and orthogonal polynomials, (eds P.B. Acosta-Humánez and F. Marcellán), Contemp. Math., Amer. Math. Soc., Providence, 509 (2010), 1-58. doi: 10.1090/conm/509.
[12]	D. Blázquez-Sanz and Ch. Pantazi, A note on the Darboux theory of integrability of non-autonomous polynomial differential systems, Nonlinearity, 25 (2012), 2615-2624. doi: 10.1088/0951-7715/25/9/2615.
[13]	M. Carnicer, The Poincaré problem in the nondicritical case, Ann. Math., 140 (1994), 289-294. doi: 10.2307/2118601.
[14]	G. Casale, Feuilletages singuliers de codimension un, groupoide de Galois et intègrales premieres, Ann. Inst. Fourier, 56 (2006), 735-779. doi: 10.5802/aif.2198.
[15]	D. Cerveau and A. Lins Neto, Holomorphic foliations in $\mathbbC \mathbbP (2)$ having an invariant algebraic curve, Ann. Inst. Fourier, 41 (1991), 883-903. doi: 10.5802/aif.1278.
[16]	E. S. Cheb-Terrab, Solutions for the general, confluent and biconfluent Heun equations and their connection with Abel equations, J. Phys. A, 37 (2004), 9923-9949. doi: 10.1088/0305-4470/37/42/007.
[17]	E. S. Cheb-Terrab and A. D. Roche, An Abel ordinary differential equation class generalizing known integrable classes, European J. Appl. Math., 14 (2003), 217-229. doi: 10.1017/S0956792503005114.
[18]	T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978.
[19]	C. Christopher, Invariant algebraic curves and conditions for a center, Proc. Roy.Soc. Edinburgh Sect. A, 124 (1994), 1209-1229. doi: 10.1017/S0308210500030213.
[20]	C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Darboux integrating factors: Inverse problems, J. Diff. Equat., 250 (2011), 1-25. doi: 10.1016/j.jde.2010.10.013.
[21]	C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Inverse problems for multiple invariant curves, Proc. Roy.Soc. Edinburgh Sect. A, 137 (2007), 1197-1226. doi: 10.1017/S0308210506000400.
[22]	C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Inverse problems for invariant algebraic curves: Explicit computations, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 287-302. doi: 10.1017/S0308210507001175.
[23]	C. Christopher, J. Llibre, Ch. Pantazi and X. Zhang, Darboux integrability and invariant algebraic curves for planar polynomial systems, J. Phys. A, 35 (2002), 2457-2476. doi: 10.1088/0305-4470/35/10/310.
[24]	C. Christopher, J. Llibre and J. V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math., 229 (2007), 63-117. doi: 10.2140/pjm.2007.229.63.
[25]	G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. math. 2ème série, 2 (1878), 60-96; 123-144; 151-200.
[26]	V. A. Dobrovol'skii, N. V. Lokot and J.-M. Strelcyn, Mikhail Nikolaevich Lagutinkskii (1871-1915): un mathématicien méconnu, Historia Math., 25 (1998), 245-264. doi: 10.1006/hmat.1998.2194.
[27]	A. Duval and M. Loday-Richaud, Kovacic's algorithm and its application to some families of special functions, Appl. Algebra Engrg. Comm. Comput., 3 (1992), 211-246. doi: 10.1007/BF01268661.
[28]	G. H. Halphen, Traité des Fonctions Elliptiques, Vol. I, II, Gauthier-Villars, Paris, 1888.
[29]	I. A. García, H. Giacomini and J. Giné, Generalized nonlinear superposition principles for polynomial planar vector fields, J. Lie Theory, 15 (2005), 89-104.
[30]	H. Giacomini , J. Giné and M. Grau, Integrability of planar polynomial differential systems through linear differential equations, Rocky Mountain J. Math., 36 (2006), 457-485. doi: 10.1216/rmjm/1181069462.
[31]	E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944.
[32]	J. P. Jouanolou, Equations de Pfaff Algébriques, Lectures Notes in Mathematics, 708, Springer-Verlag, New York/Berlin, 1979. doi: 10.1007/BFb0063393.
[33]	I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1957.
[34]	T. Kimura, On Riemann's equations which are solvable by quadratures, Funkcialaj Ekvacioj, 12 (1969), 269-281.
[35]	J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symb. Comput., 2 (1986), 3-43. doi: 10.1016/S0747-7171(86)80010-4.
[36]	E. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, Pure and Applied Mathematics, 54, New York - London, 1973.
[37]	A. Lins Neto, Construction of singular holomorphic vector fields and foliations in dimension two, J. Differential Geometry, 26 (1987), 1-31.
[38]	J. Llibre and Ch. Pantazi, Polynomial differential systems having a given Darbouxian first integral, Bull. Sci. Math, 128 (2004), 775-788. doi: 10.1016/j.bulsci.2004.04.001.
[39]	J. Llibre and Ch. Pantazi, Darboux theory of integrability for a class of nonautonomous vector fields, J. Math. Phys., 50 (2009), 102705, 19pp. doi: 10.1063/1.3205450.
[40]	J. Llibre, Ch. Pantazi and S. Walcher, Morphisms and inverse problems for Darboux integrating factors, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1291-1302. doi: 10.1017/S0308210511001430.
[41]	J. Llibre and C. Valls, Analytic integrability of quadratic-linear polynomial differential systems, Ergod. Th. & Dynam. Sys., 31 (2011), 245-258. doi: 10.1017/S0143385709000868.
[42]	J. Llibre and X. Zhang, Darboux theory of integrability in $\C^n$ taking into account the multiplicity, J. Diff. Equat., 246 (2009), 541-551. doi: 10.1016/j.jde.2008.07.020.
[43]	J. Llibre and X. Zhang, Darboux theory of integrability for polynomial vector fields in $\mathbb R^n$ taking into account the multiplicity at infinity, Bull. Sci. Math., 133 (2009), 765-778. doi: 10.1016/j.bulsci.2009.06.002.
[44]	B. Malgrange, Le groupoide de Galois d'un feuilletage, in Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., 38 (2001), 465-501.
[45]	B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, 23 (2002), 219-226. doi: 10.1142/S0252959902000213.
[46]	J. Martinet and J. P. Ramis, Theorie de Galois differentielle et resummation, in Computer Algebra and Differential Equations, (ed. E. Tournier), Comput. Math. Appl., Academic Press, London, (1990), 117-214.
[47]	J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems, Progress in Mathematics, 179, Birkhäuser, Basel, 1999. doi: 10.1007/978-3-0348-8718-2.
[48]	Ch. Pantazi, Inverse Problems of the Darboux Theory of Integrability for Planar Polynomial Differential Systems, PhD. thesis, Universitat Autonoma de Barcelona, 2004.
[49]	J. V. Pereira, Vector fields, invariant varieties and linear systems, Annales de l'institut Fourier, 51 (2001), 1385-1405. doi: 10.5802/aif.1858.
[50]	J. V. Pereira, On the Poincaré problem for foliations of general type, Math. Ann. , 323 (2002), 217-226. doi: 10.1007/s002080100277.
[51]	A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Second Edition, Chapman and Hall, Boca Raton, 2003.
[52]	E. G. C. Poole, Introduction to the Theory of Linear Differential Equations, Dover Publications, Inc., New York, 1960.
[53]	M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc., 279 (1983), 215-229. doi: 10.1090/S0002-9947-1983-0704611-X.
[54]	M. van der Put and M. Singer, Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer Verlag, Berlin, 2003. doi: 10.1007/978-3-642-55750-7.
[55]	M. F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc., 333 (1992), 673-688. doi: 10.1090/S0002-9947-1992-1062869-X.
[56]	M. F. Singer, Liouvillian solutions of $n$th order homogeneous linear differential equations, Amer. J. Math., {103} (1981), 661-682. doi: 10.2307/2374045.
[57]	F. Ulmer and J. A. Weil., Note on Kovacic's algorithm, J. Symb. Comp., 22 (1996), 179-200. doi: 10.1006/jsco.1996.0047.
[58]	R. Vidunas, Differential equations of order two with one singular point, J. Symb. Comp., 28 (1999), 495-520. doi: 10.1006/jsco.1999.0312.
[59]	J.-A. Weil, Constantes et Polynômes de Darboux en Algèbre Differentielle: Applications Aux Systèmes Différentiels Linéaires, PhD. Thesis, École Politechnique, 1995.
[60]	J. A. Weil, Recent algorithms for solving second-order differential equations, in Algorithms Seminar 2001-2002, (ed. F. Chyzak), INRIA, (2003), 43-46.
[61]	E. T. Whittaker and E. T. Watson, A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions: With an Account of the Principal Transcendental Functions, Cambridge Univ. Press, New York, 1962.
[62]	H. Zołądek, Polynomial Riccati equations with algebraic solutions, in Differential Galois theory (Bedlewo, 2001), Banach Center Publ., 58, Polish Acad. Sci., Warsaw, (2002), 219-231. doi: 10.4064/bc58-0-17.

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