  We consider gauge theories on Poisson manifolds emerging as semiclassical
approximations of noncommutative spacetime with Lie algebra type
noncommutativity. We prove an important identity, which allows to obtain simple
and manifestly gauge-covariant expressions for the Euler-Lagrange equations of
motion, the Bianchi and the Noether identities. We discuss the non-Lagrangian
equations of motion, and apply our findings to the $\kappa$-Minkowski case. We
construct a family of exact solutions of the deformed Maxwell equations in the
vacuum. In the classical limit, these solutions recover plane waves with
left-handed and right-handed circular polarization, being classical
counterparts of photons. The deformed dispersion relation appears to be
nontrivial.
