  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
Painlev\'e equations (i.e., equations in the polynomial class having the
Painlev\'e property). Like the classical Painlev\'e equations, this equation
admits a Hamiltonian formulation, B\"acklund 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.
