B\"{a}cklund Transformations of Einstein's Field Equations for the Interior of a Uniformly Rotating Stationary Axisymmetric Perfect Fluid

Clairin's method of obtaining B\"{a}cklund transformations is applied to Einstein's field equations for the interior of a uniformly rotating stationary axisymmetric perfect fluid. It is shown that for arbitrary pressure $p$ and mass density $\mu$ the method does not give non-trivial B\"{a}cklund transformations, while if $\mu + 3p =0$ it gives the transformation of Ehlers.