Summary In my first review  of van Vlaenderen's article  I stated that his theory is not compatible with the Maxwell theory. Van Vlaenderen replied in  that he intended a modification of the Maxwell equations consciously. Therefore in the following this modification is checked. It turns out that van Vlaenderen modifies the inhomogenities of two Maxwell equations, the charge density ρ and the current density J. We obtain the following result: While ρ and J - as is well-known - fulfil the conservation law of charges, the same is not true for van Vlaenderen's modifications, which makes the physical relevance of van Vlaenderen's theory questionable.
As can be seen from van Vlaenderen's response  to my web article  his theory consists in a modification of the well-known Maxwell equations of electrodynamics. His basic equations are
−ÑΦ − At = E the electric field (1)
Ñ×A = B the magnetic field (2)
which are equivalent to the "homogeneous" Maxwell equations
(1') curl E + Bt= 0 ,
(2') div B = 0 ,
and the inhomogenious differential equations
1/c² Φtt − ΔΦ = ρ/εo (4)
1/c² Att − ΔA = μoJ (5)
where 1/c² = εoμo. Due to (1) and (4) we obtain
(4') div E = 1/c² Φtt − ΔΦ + St = ρ/εo + St
where S denotes van Vlaenderen's "seventh field component" given by
λoεoμo ∂Φ/∂t − Ñ·A = S the scalar field, a seventh field component (3)
in the case λo=1 under consideration. Similarly (2) and (5) yield
(5') curl B − 1/c² Et = 1/c² Att − ΔA + St = μoJ − grad S
The red marked terms in (4'-5') show the deviations from the corresponding inhomogeneous Maxwell equations if a non-constant function S is used.
The deviations mean that the local charge ρ and the current J of the charge are modified: Instead of the true charge and current van Vlaenderen uses modified charge and current fields:
(6') ρ' := ρ + εoSt , J' := J − 1/μograd S
As is well-known charge density ρ and current density J of the Maxwell equations fulfil the conservation law for charges
(7') ρt + div J = 0 .
The check for the conservation law of van Vlaenderen's modified charge and current fields yields
(8') ρ't + div J' = εoStt − 1/μoΔS = 1/μo (1/c²Stt − ΔS) .
Thus we have the results:
The result (I) is a strong argument against the physical validity of van Vlaenderen's modification of the Maxwell equations. Maybe that some reasons can be found for the validity of condition (II).
Remark Van Vlaenderen has repeatedly remarked that he thinks the usual gauge theory to be of circular logic. I cannot follow his ideas and refer to the literature, e.g. to my article . If necessary this topic could be discussed in another article.
Koen van Vlaenderen: Generalised Classical Electrodynamics
for the prediction of scalar field effects
Gerhard W. Bruhn: A Remark on van Vlaenderen's Seventh Field Component
Koen van Vlaenderen: Gerhard Bruhn's “Remark” on
my scalar field theory is wrong
Gerhard W. Bruhn: Gauge Theory of the Maxwell Equations
 Ernst Schmutzer, Grundlagen der Theoretischen Physik, Teil 1 p.588 ff.