Non-linear electrodynamics generalization of the rotating BTZ black hole

We obtain two exact solutions of Einstein gravity coupled to nonlinear electrodynamics (NLED) in $(2+1)$-dimensional Anti-de Sitter (AdS) spacetime. The solutions are characterized by the mass $M$, angular momentum $J$, cosmological constant or (anti) de Sitter parameter $\mathrm{\Lambda }$, and an electromagnetic parameter $Q$, that is related to an electric field in the first solution and to a magnetic charge for the second solution. Depending on the range of the parameters, the solutions admit a charged rotating asymptotically AdS black hole (BH) interpretation or a charged rotating asymptotically AdS traversable wormhole (WH). If the electromagnetic field is turned off, the stationary Ba~nados-Teitelboim-Zanelli (BTZ) BH is recovered; in such a way that our BH-WH solutions are nonlinear charged generalizations of the stationary BTZ-BH. Moreover, in contrast to the BTZ metric, the derived AdS solutions are singular at certain radius $r_s\mathrm{\ne }0$, resembling the ring singularity of the Kerr-Newman spacetime; while if $\mathrm{\Lambda }$ is positive the curvature invariants of the second solution are finite.