What are the Lorentz Transformations between polar coordinates? Or can Lorentz Transformations be Non-Linear?

This question rises from the comments on @G Smith’s answer’s to this question https://physics.stackexchange.com/a/603032/113699

Precisely I was trying to understand the Lorentz Transformations between Spherical Polar Coordinates. The point is that I have read that Lorentz transformations have to be Linear but if for spherical polar coordinates they will be Non-Linear.

These answers here say that Interval preserving transformations/ Lorentz Transformations need to be linear
Interval preserving transformations are linear in special relativity and Kleppner derivation of Lorentz transformation ( the answer by Selena Routley) and Why is this non-linear transformation not a Lorentz transformations? It does preserve $x^2 + y^2 + z^2 – c^2t^2 = x'^2 + y'^2 + z'^2 – c^2t'^2 $ ( especially Knzhou’s comment on A Hussain’s answer where he also says we should do SR in cartesian coordinates and not polar coordinates).

But the answer by Void here Are Lorentz transformations linear transformations?

says that Lorentz Transformations should be Homogenous and not Linear.

So my question is–

Can be there Lorentz Transformations between spherical polar coordinates or cylindrical coordinates. If yes, then they will be Non-Linear. So can the Lorentz Transformations be Non-Linear too?