  In this paper, in the frame of Extended Electrodynamics (EED), we study some
of the consequences that can be obtained from the introduced and used by
Maxwell equations complex structure \mathcal{J} in the space of 2-forms on
\mathbb{R}^4, and also used in EED. First we give the vacuum EED equations with
some comments. Then we recall some facts about the invariance group $G$ (with
Lie algebra \mathcal{G}) of the standard complex structure $J$ in \mathbb{R}^2.
After defining and briefly studying a representation of $G$ in the space of
2-forms on \mathbb{R}^4 and the joint action of $G$ in the space of
\mathcal{G}-valued 2-forms on \mathbb{R}^4 we consider its connection with the
vacuum solutions of EED. Finally we consider the case with point dependent
group parameters and show that the set of the nonlinear vacuum EED solutions is
a disjoint union of orbits of the $G$-action, noting some similarities with the
quantim mechanical eigen picture and with the QFT creation and anihilation
operators.
