  We study a Sturm-Liouville type eigenvalue problem for second-order
differential equations on the infinite interval. Here the eigenfunctions are
nonzero solutions exponentially decaying at infinity. We prove that at any
discrete eigenvalue the differential equations are integrable in the setting of
differential Galois theory under general assumptions. Our result is illustrated
with two examples for a stationary Schroedinger equation having a generalized
Hulthen potential and an eigenvalue problem for a traveling front in the
Allen-Cahn equation.
