Truncation-type methods and Backlund transformations for ordinary differential equations: the third and fifth Painleve equations

In a recent paper we presented a truncation-type method of deriving Backlund transformations for ordinary differential equations. This method is based on a consideration of truncation as a mapping that preserves the locations of a natural subset of the movable poles that the equation possesses. Here we apply this approach to the third and fifth Painleve equations. For the third Painleve equation we are able to obtain all fundamental Backlund transformations for the case where the parameters satisfy $\gamma \delta \mathrm{\ne }0$. For the fifth Painleve equation our approach yields what appears to be all known Backlund transformations.