Since Kähler action is invariant also under ordinary gauge transformations one can formally derive the analog of conserved gauge charge for non-constant gauge transformation Φ. The question is whether this current has any physical meaning.
One obtains current as contraction of Kähler form and gradient of Φ:
j^{α}_{Φ}= J^{αβ}∂_{β}Φ ,
which is conserved only if Kähler current vanishes so that Maxwell's equations are true or if the contraction of Kähler current with gradient of Φ vanishes:
j^{α}_{Φ}∂_{α}Φ=0 .
The construction of preferred extremals leads to the proposal that the flow lines of Kähler current are integrable in the sense that one can assign a global coordinate Ψ with them. This means that Kähler current is
proportional to gradient of Ψ:
j^{α}_{Φ}= g^{αβ} ∂_{β}Ψ .
This implies that the gradients of Φ and Ψ are orthogonal. If Kähler current is light-like as it is for the known extremals, Φ is superposition of light-like gradient of Ψ and of two gradients in a sub-space of tangent space analogous to space of two physical polarizations. Essentially the local variant of the polarization-wave vector geometry of the modes of radiative solutions of Maxwell's equations is obtained. What is however important that superposition is possible only for modes with the same local direction of wave vector (∇Ψ) and local polarization.
Kähler current would be scalar function k times gradient of Ψ :
j^{α}_{Φ}= kg^{αβ}∂_{β}Ψ .
The proposal for preferred extremals generalizing at least MEs leads to the proposal that the extremals define two light-like coordinates and two transversal coordinates.