Jefimenko, for example, got the answer by solving the fields generated by a uniformly moving charged explicitly. This can be done and is not too difficult. However, Maxwell's equations are covariant under Lorentz transformations. It is thus also possible to obtain this solution by transforming using the Lorentz transformation the electrostatic field. Both ways are correct ways of solving the Maxwell equations, hence both give the same solutions, since the outgoing wave solution of Maxwell's equations with given sources is unique.

Neither approach, however, can lay claim to being ``Galilean''. Maxwell's equations are not invariant under the Galilei group, nor has the solution in any way been derived using Galilei invariance.

Dear Leyvraz

Your contribution was of great value, if only because, in a sense, it may invalidate my question. However, what we have here is just a matter of semantics (or something like that). You alerted me to the fact that I should have mentioned in the comments that in the solution of a physical problem we always have a coordinate system. If no mention is made in relation to the coordinate system, then it will be the one that emanates from our senses (empirical); commonly called Galilean, in which the time evolves regardless of the coordinates. Better, the internal clock of moving bodies doesn’t delay with respect to stationary observers (contradicting the Hafele-Keating experiment).

In short, I agree with you in the sense that a coordinate system is characterized by the type of operations that it allows; coordinate transformations. Poincaré showed in first approximation that the laws of physics should be covariant, otherwise describe different phenomena in different reference frames. Lorentz took this seriously (principle of relativity) and discovered that Maxwell's equations are covariant only under Lorentz transformations (before Einstein). On the other hand, we must remember that the Schrödinger equation, although not covariant in any of the systems mentioned above, can provide the wonderful Hydrogen Atom solution (Extensible to other atoms). So, covariance must be better understood; especially taking into account that the inertia appears due to movement through an interacting vacuum (Planck’s vacuum).

Ok, let us return to the question whether Galilean transformation of coordinates is implicit or not (or is irrelevant to mention it) in the derivation of the electromagnetic fields of a moving charge.

For example, Jefimenko used the retarded technique (Liénard-Wiechert 1898) in a fixed frame (so, no problems with the light speed). However, it is introduced a time delay (retardation) connecting two positions via speed of light. One is the actual position of the charge (moving) and in the other what is seen, from the observation point, is the instant effect of the charge in question, as if it were in this advanced position. Inexorably, we are faced with a coordinate transformation (not canonical, but inherent), which is Galilean.

All other derivations, concerning free electrical charges, one way or another, involve transformations of Galileo.

Once solved the relativistic effect, the relative motion of bodies (or electrical charges) occurs according to the Galilean transformations. This is proven in the references:

Schwinger J., DeRaad L.L., Milton K.A. Tsai W. Classical Electrodynamics. Perseus Books, 1998. page 111.

Daywitt W.C. The Lorentz Transformation as a Planck Vacuum Phenomenon in a Galilean Coordinate System. Progress in Physics, 2011, v. 2, 3-6.