This paper can be viewed as a random walk through the subject of nilpotent and globally nilpotent connections. The author first deduces from Katz' theorem and his own work [Am. J. Math. 112, 749-786 (1990; Zbl 0718.12007)] that when written in the standard Fuchsian way, the coefficients of the polynomials occurring in a globally nilpotent differential operator \(L\) are integral over the ring generated over \(\mathbb{Q}\) by the singularities of \(L\). He applies this result to Lamé equations, thus recovering several classical results on their accessory parameter. He then proceeds to the case of systems, where no such integrability result is known. Using the transcendency of some of its `accessory parameters', he shows that the generic system with logarithmic poles at 0 and 1 and residues of a certain type is not globally nilpotent. The proof goes through a reduction to differential equations.