Strings in N=2 supersymmetric U(1)N gauge theories with N hypermultiplets are studied in the generic setting of an arbitrary Fayet-Iliopoulos triplet of parameters for each gauge group and an invertible charge matrix. Although the string tension is generically of a square-root form, it turns out that all existing Bogomol'nyi-Prasad-Sommerfield solutions have a tension that is linear in the magnetic fluxes, which, in turn, are linearly related to the winding numbers. The main result is a series of theorems establishing three different kinds of solutions of the so-called constraint equations, which can be pictured as orthogonal directions to the magnetic flux in SU(2)R space. We further prove for all cases that a seemingly vanishing Bogomol'nyi bound cannot have solutions. Finally, we write down the most general vortex equations in both master form and Taubes-like form. Remarkably, the final vortex equations essentially look Abelian in the sense that there is no trace of the SU(2)R symmetry in the equations after the constraint equations have been solved.
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