Movable singular point

A singular point $z_0$ of the solution $w(z)$ of a differential equation $F(z,w,w')=0$ ($F$ is an analytic function), where $w(z)$ is considered as a function of the complex variable $z$, which is such that solutions to the same equation with initial data close to the original data have singular points close to $z_0$ but not coincident with it. The classical example of a movable singular point arises when considering the equation

$$\frac{dw}{dz}=\frac{P(z,w)}{Q(z,w)},$$

where $P$ and $Q$ are holomorphic functions in a certain region of the space $\mathbf C^2$. If the surface $\{Q=0\}$ is irreducible and is projected along the $Ow$-axis on a region $\Omega\subset Oz$, then all points in the region $\Omega$ are movable singular points; for the solution with initial condition $(z_0,w_0)$, where

$$Q(z_0,w_0)=0\neq P(z_0,w_0),$$

the point $z_0$ is an algebraic branch point.

References

[1]	V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) MR0100119

Comments

For equations of the form

$$\frac{d^2w}{dz^2}=R\left(\frac{dw}{dz},w,z\right),$$

where $R$ is rational in $dw/dz$ and $w$ and analytic in $z$, it is known which equations have only non-movable singularities, cf. Painlevé equation and [a1].

References

[a1]	E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) MR1570308 MR0010757 MR1524980 MR0000325 MR1522581 Zbl 0612.34002 Zbl 0191.09801 Zbl 0063.02971 Zbl 0022.13601 Zbl 65.1253.02 Zbl 53.0399.07