The authors discuss Bäcklund transformations and solution hierarchies for the third Painlevé equation $\text{P}_{\text{III}}$: $$y''={1\over y} (y')^2- {1\over x} y'+{1\over x} (\alpha y^2+\beta)+\gamma y^3+{\delta\over y}, \Biggl('={d\over dx}\Biggr),\tag1$$ where $\alpha$, $\beta$, $\gamma$ and $\delta$ are arbitrary constants.
A survey of the study of Painlevé equations is given in Section 1. The integration of the continuous $\text{P}_{\text{III}}$ with $\beta=\delta=0$ or $\alpha=\gamma=0$ and several other properties are reviewed in Section 2, and many scaling transformations are also shown therein. In Section 3, various Bäcklund type transformations for $\text{P}_{\text{III}}$ are described. Section 4 is devoted to the parameter sets for which exact solutions of $\text{P}_{\text{III}}$ exist. In Section 5, these exact solutions are categorized into three hierarchies: solutions rational in $x$; solutions can be expressed by Bessel functions; and solutions rational in $x^{1/3}$.
In Section 5, the following discrete analogy of $\text{P}_{\text{III}}$ (d-P$_{\text{III}}$) $$y_{n+1}y_n= {\nu y^2_n-\beta x_ny_n-\delta x^2_n\over\gamma x^2_ny^2_n+\alpha x_ny_n+\nu}\tag2$$ is considered. The rational solution of (2) with $\gamma\delta\ne 0$ and exact solutions of (2) with $\gamma$ and $\delta$ being zero are considered. A final conclusion is stated in Section 6.
This long paper is interesting because the third Painlevé equation has a large number of physically significant applications. A bibliography of 78 papers is included.