  Robinson--Trautman solutions with Nonlinear Electrodynamics are investigated
for both L(F ) and L(F, G) Lagrangians and presence of electric and magnetic
charges as well as electromagnetic radiation is assumed. Particular interest is
devoted to models representing regular black holes for spherically symmetric
situations. The results show clear uniqueness of Maxwell electrodynamics with
respect to compatibility with Robinson--Trautman class. Additionally, regular
black hole models are clearly not suited to this class while famous
Born--Infeld model illustrates important distinction between L(F ) and L(F, G)
for obtained electric field when magnetic field is nontrivial.
