  In the present work, we study four-dimensional black strings in Horndeski
models with translation invariance. Imposing that the scalar field depends on
the string-generator coordinate, the Klein-Gordon equation admits a linear
profile as a solution. This relaxation allows finding rotating, asymptotically
AdS$_3\times \mathbb{R}$ black strings, dressed with an effective cosmological
constant. In this regard, we show that in the full spectrum of shift-symmetric
Horndeski theories with Einstein limit, the scalar charge needs to be fixed in
terms of the parameter space. This method is employed to concrete examples to
illustrate the scheme we go along with. Regarding the conserved charges, we
compute them via the Euclidean method and show the fulfillment of the
associated Smarr Law. Finally, we exhibit that our AdS strings are locally and
globally stable under small fluctuations around the equilibrium.
