The Rotating Tautochrone

In a recent paper by Flores and Osler, the authors investigated tautochrone curves in the xy-plane under an arbitrary potential $Vy.$ In this paper we imagine that the xy-plane of the tautochrone curve is rotating about the y-axis with constant angular momentum. We find the differential equation of the tautochrone curves. While this differential equation is difficult to solve analytically, several exact solutions (in terms of elementary functions) are obtained in an indirect manner. Intuitive motivation for tautochrone motion is given.