Buchdahl-like transformations for perfect fluid spheres

In two previous articles [Phys. Rev. D71 (2005) 124307 (gr-qc/0503007), and gr-qc/0607001] we have discussed several "algorithmic" techniques that permit one (in a purely mechanical way) to generate large classes of general relativistic static perfect fluid spheres. Working in Schwarzschild curvature coordinates, we used these algorithmic ideas to prove several "solution-generating theorems" of varying levels of complexity. In the present article we consider the situation in other coordinate systems: In particular, in general diagonal coordinates we shall generalize our previous theorems, in isotropic coordinates we shall encounter a variant of the so-called "Buchdahl transformation", while in other coordinate systems (such as Gaussian polar coordinates, Synge isothermal coordinates, and Buchdahl coordinates) we shall find a number of more complex "Buchdahl-like transformations" and "solution-generating theorems" that may be used to investigate and classify the general relativistic static perfect fluid sphere. Finally by returning to general diagonal coordinates and making a suitable ansatz for the functional form of the metric components we place the Buchdahl transformation in its most general possible setting.