Summary: We prove the invertibility of the modified Stokes-type operator \(\Xi +W\) on a manifold with straight cylindrical ends, where \[ \Xi := \begin{pmatrix} 2\operatorname{Def}^{\ast} \operatorname{Def} & d \\ d^{\ast} & 0 \end{pmatrix} \text{ and } W := \begin{pmatrix} V & 0 \\ 0 & - V_0 \end{pmatrix} \] are the usual Stokes operator and a suitable vector potential, respectively. The proof is based on the ‘Mitrea-Taylor trick.’ To be able to use this method, we rely on some auxiliary Fredholm, regularity, and invertibility results, some established in this paper and some in a recent preprint [M. Kohr et al., “Layer potentials and essentially translation invariant pseudodifferential operators on manifolds with cylindrical ends”, Preprint, arXiv:2308.06308]. The auxiliary results are proved using a suitable pseudodifferential calculus on our manifold with cylindrical ends. As a consequence, we also obtain the structure of the inverse of our modified Stokes operator. Our main motivation for these results is the study of layer potential operators for the Stokes system on domains with cylindrical ends (or outlets). For this reason, we formulate many of our results in the framework of Douglis-Nirenberg-elliptic operators.