Abstract
Every static traversable wormhole reported so far in
$$(2+1)$$
(
2
+
1
)
gravity coupled to nonlinear electrodynamics has been obtained from a Lagrangian of the power-Maxwell form
$$\mathcal {L}\propto |F|^{k}$$
L
∝
|
F
|
k
with the single value
$$k=1/2$$
k
=
1
/
2
, and in every case with a cosmological constant forced to a fixed sign or to zero; the same Lagrangian, again with
$$\varLambda <0$$
Λ
<
0
, underlies the known stationary family. We show that, within the static sector, these restrictions are artefacts of that one choice of Lagrangian, not physical requirements, and give what we argue is a complete classification of static traversable wormholes in this theory. The stationary sector is not addressed here and is not a special case of the present analysis. Of the three mutually exclusive electromagnetic configurations compatible with the symmetry, the radial electric branch admits no throat for any Lagrangian or cosmological constant. In each of the remaining two branches, azimuthal electric and magnetic, the field equations leave only two possibilities: either the redshift function is fixed (
$$e^{\varPhi }=Cr$$
e
Φ
=
C
r
, or
$$\varPhi =\text {const}$$
Φ
=
const
, respectively), in which case the shape function
b
(
r
) is completely free and
$$\varLambda $$
Λ
is absorbed without constraint; or it is not, in which case
b
(
r
) becomes the free function instead, with the redshift determined from it. We solve both regimes in closed or explicit form in both branches, and prove that fixing
$$\mathcal {L}(F)$$
L
(
F
)
to any single power
k
, not only
$$k=1/2$$
k
=
1
/
2
, leaves the shape function determined up to a finite number of integration constants, and in the azimuthal branch forces
$$\varLambda \ne 0$$
Λ
≠
0
: the freedom found here requires
$$\mathcal {L}(F)$$
L
(
F
)
to be genuinely unrestricted. Along the way we identify an azimuthal-electric family whose shape function is exactly the static BTZ mass function, the same geometry that is a black hole in vacuum becoming a traversable wormhole once sourced by the nonlinear field instead. Finally, we apply the classification to the unique conformally invariant power-Maxwell theory in
$$(2+1)$$
(
2
+
1
)
dimensions,
$$k=3/4$$
k
=
3
/
4
, previously known only to source a Coulomb-like charged black hole in the radial branch: the same Lagrangian produces genuine traversable wormholes in the azimuthal and magnetic branches, showing that it is the electromagnetic configuration, not the choice of Lagrangian, that decides between a horizon and a throat; Born–Infeld electrodynamics, by contrast, never violates the null energy condition in any branch and admits no throat at all.