  We study solutions in non-linear electrodynamics (NED) and establish several
general results. We show, that the $SO(2)$ electric-magnetic duality symmetry
is restrictive enough to allow for reconstruction of the NED Lagrangian from
the spherically-symmetric electrostatic (Coulomb-like) solution -- although
there are infinitely many different NED theories admitting a given solution,
there exists a unique $SO(2)$ invariant one among them. We introduce a general
algorithm for constructing new $SO(2)$ invariant NED theories in the
conventional approach, where only a few examples are available. We also show
how to derive the Lagrangian of the $SO(2)$ invariant theory admitting a given
electrostatic solution. We further show on a simple example that some NED
theories may require sources (particles) of finite (non-zero) size. Such a
non-zero size source not only regularizes the infinite energy of the point
charge but also satisfies the condition of regularity, that the electric field
is zero at the origin. The latter condition was identified earlier as necessary
and sufficient for the NED solution to generate a regular black hole via
so-called double copy construction and is also satisfied by solitons. We
propose a large class of solitonic NED solutions that give rise to regular
black holes via double copy construction and contain solutions of Maxwell and
Born-Infeld as different limits. This class of NED solutions acquires two new
properties in the limit where the corresponding regular black hole's
asymptotics becomes Minkowski: it gives rise to regular higher-spin black holes
via generalization of double copy -- ``higher-copy'' construction, and for very
short distances changes the sign of the force becoming repulsive/attractive for
opposite/similar signs of charges.
