Lorentz transformation, non linear space transformations [duplicate]

The Lorentz transformation is linear in $x,y,z,t$.

Are there are any non linear space transformations that leaves the speed of light constant in all inertial frames? The possibility being that space or matter might not transform in a simple homogeneous way.

Such a transformation which leaves the Maxwell equations invariant I imagine is unlikely.

• The question and the respective answer by Qmechanic pointed out by ACuriousMind explicitly assume that (1) there is a spacetime viewed as a (2) differential manifold equipped with (3) a metric constructed by exploiting the constant light velocity postulate. The question above does not assume all these physical hypotheses. I do not believe that it is a duplicate even if some overlap exists. Jan 9 '17 at 11:55
• Agree with Valter Moretti comments. I am thinking of non linear transformations. The possible duplicates referred to deal more with proving the linearity of the LT I think, a different subject. Jan 9 '17 at 12:08