I was recently looking into generalisations of the brachistochrone problem: for example, in this article the authors study the brachistochrone with Amontons-Coulomb friction where a bead slides along a wire from one fixed point to another under the influence of gravity and friction.

In this one the author considers the brachistochrone for a sphere which rolls without friction. It seems like in the literature no-one has studied the brachistochrone/tautochrone curves for a homogeneous sphere with slippage between sphere and surface ie. a sphere which slides and rolls with friction (I am happy to be corrected if I am wrong on this). Is there any reason why this solution would be impossible to obtain or something to do with the effect of slip being too difficult to analyze?

I note that in the first article it is possible to express the solution in terms of elementary functions but perhaps that is no longer possible for the rolling sphere problem.