  We analytically construct asymptotically AdS$_5$ black string solutions
starting from the four-dimensional domain wall black hole of arXiv:0911.4926.
It is shown that its uplift gives a black string in $d=5$ minimal gauged
supergravity, with momentum along the string. Applying instead the residual
symmetries of $N=2$, $d=4$ Fayet-Iliopoulos-gauged supergravity discovered in
arXiv:1606.05160 to the domain wall seed leads, after uplifting, to a dyonic
black string that interpolates between AdS$_5$ and
$\text{AdS}_3\times\text{H}^2$ at the horizon. A Kaluza-Klein reduction of the
latter along an angular Killing direction $\phi$ followed by a duality
transformation yields, after going back to five dimensions, a black string with
both momentum along the string and rotation along $\phi$. This is the first
instance of using solution-generating techniques in gauged supergravity to add
rotation to a given seed. These solutions all have constant scalar fields. As
was shown in hep-th/0302218, the construction of supersymmetric static magnetic
black strings in the FI-gauged stu model amounts to solving the
$\text{SO}(2,1)$ spinning top equations, which descend from an inhomogeneous
version of the Nahm equations. We are able to solve these in a particular case,
which leads to a generalization of the Maldacena-Nu\~nez solution.
