*(English)*Zbl 0846.34002

The author determines all “classical” solutions (in the sense of Painlevé and Umemura) of the third Painlevé equation. The equation is equivalent to

After showing that ${P}_{II{I}^{\text{'}}}$ has a rational solution if and only if $\alpha =\gamma =0$ or $\beta =\delta =0$ and reviewing a result of V. I. Gromak for the case of $\gamma =0$, $\alpha \delta \ne 0$ (or $\delta =0$, $\beta \gamma \ne 0$), the author studies the case of $\gamma \delta \ne 0$ which is equivalent to the case of $\alpha =-4{\theta}_{\infty}$, $\beta =4({\theta}_{0}+1)$, $\gamma =4$, $\delta =-4$. The equation ${P}_{II{I}^{\text{'}}}$ with the above parameters is denoted by ${P}_{II{I}^{\text{'}}}({\theta}_{0},{\theta}_{\infty})$.

The main results of this paper are stated as follows: (1) ${P}_{II{I}^{\text{'}}}({\theta}_{0},{\theta}_{\infty})$ does not have rational solutions, (2) ${P}_{II{I}^{\text{'}}}({\theta}_{0},{\theta}_{\infty})$ has algebraic solutions if and only if ${\theta}_{\infty}-{\theta}_{0}=1$ or ${\theta}_{\infty}+{\theta}_{0}+1$ is an even integer, (3) the number of algebraic solutions (in statement (2)) are one or two, and the latter occurs if and only if both ${\theta}_{\infty}-{\theta}_{0}-1$ and ${\theta}_{\infty}+{\theta}_{0}+1$ are even integers, (4) if ${\theta}_{\infty}+{\theta}_{0}$ (or ${\theta}_{\infty}-{\theta}_{0})$ is an even integer, then ${P}_{II{I}^{\text{'}}}({\theta}_{0},{\theta}_{\infty})$ has a one-parameter family of classical solutions which is a rational function of ${\theta}_{0}$, $t$ and a general solution of a Riccati equation $dq/dt=-{q}^{2}/t-{\theta}_{0}q/t+1$. Furthermore, the exact forms of algebraic solutions are given.

The author announces that the irreducibility of ${P}_{II{I}^{\text{'}}}$ except for the classical solutions in this paper will be proved in his forthcoming paper.