Painlevé property

Remarks

For the first, second and fourth Painlevé equations, the only fixed singular point is $\infty$, and every solution is a meromorphic function on all of $\C$. For the other three, solutions can still branch at the fixed singular points, such as $0$, $1$ and $\infty$ for the sixth equation. Some authors only require that no movable singularity is a branch point.

Example: the Riccati equation $y' = y^2$ has the solutions $y = 0$ and $y = \frac{1}{c - z}$, $c \in \C$. The only singularity is the simple pole at $z = c$, which moves with $c$, so this movable singularity is a pole. More generally, every Riccati equation $y' = a(z) + b(z) y + c(z) y^2$ has the Painlevé property.

Non-example: $y' = y^3$ has the solutions $y^2 = \frac{1}{2 (c - z)}$, so $y = \pm \left( 2 (c - z) \right)^{-1/2}$. The point $z = c$ is a movable branch point and not a pole, so the property fails.

The first Painlevé equation $y'' = 6 y^2 + z$ has the Painlevé property, and its solutions are not expressible by classical functions in general.