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Strings in $\mathcal{N}=2$ supersymmetric ${\rm U}(1)^N$ gauge theories with$N$ hypermultiplets are studied in the generic setting of an arbitraryFayet-Iliopoulos triplet of parameters for each gauge group and an invertiblecharge matrix. Although the string tension is generically of a square-rootform, it turns out that all existing BPS (Bogomol'nyi-Prasad-Sommerfield)solutions have a tension which is linear in the magnetic fluxes, which in turnare linearly related to the winding numbers. The main result is a series oftheorems establishing three different kinds of solutions of the so-calledconstraint equations, which can be pictured as orthogonal directions to themagnetic flux in ${\rm SU}(2)_R$ space. We further prove for all cases, that aseemingly vanishing Bogomol'nyi bound cannot have solutions. Finally, we writedown the most general vortex equations in both master form and Taubes-likeform. Remarkably, the final vortex equations essentially look Abelian in thesense that there is no trace of the ${\rm SU}(2)_R$ symmetry in the equations,after the constraint equations have been solved.

1/2-BPS vortex strings in $\mathcal{N}=2$ supersymmetric ${\rm U}(1)^N$ gauge theories