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Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we study the equation $$ w^{(4)} = 5 w'' (w^2 - w') + 5 w (w')^2 - w^5 + (\lambda z + \alpha)w + \gamma, $$ which is one of the higher-order Painlevé equations (i.e., equations in the polynomial class having the Painlevé property). Like the classical Painlevé equations, this equation admits a Hamiltonian formulation, Bäcklund transformations and families of rational and special functions. We prove that this equation considered as a Hamiltonian system with parameters $\gamma/\lambda = 3 k$, $\gamma/\lambda = 3 k - 1$, $k \in \mathbb{Z}$, is not integrable in Liouville sense by means of rational first integrals. To do that we use the Ziglin–Morales-Ruiz–Ramis approach. Then we study the integrability of the second and third members of the $\mathrm{P}_{\mathrm{II}}$-hierarchy. Again as in the previous case it turns out that the normal variational equations are particular cases of the generalized confluent hypergeometric equations whose differential Galois groups are non-commutative and hence, they are obstructions to integrability.
Keywords: Painlevé type equations; Hamiltonian systems; differential Galois groups; generalized confluent hypergeometric equations. 
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     author = {Ognyan Christov and Georgi Georgiev},
     title = {Non-Integrability of {Some} {Higher-Order} {Painlev\'e} {Equations} {in~the~Sense} {of~Liouville}},
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Ognyan Christov; Georgi Georgiev. Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a44/

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