Non-Integrability of Some Higher-Order Painleve Equations in the Sense of Liouville

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 Painleve equations (i.e., equations in the polynomial class having the Painleve property). Like the classical Painleve equations, this equation admits a Hamiltonian formulation, Backlund 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.