Given an ordinary differential equation (ODE) for an analytic function from a domain in the complex plane (or on a Riemann surface) to the complex numbers, a point is a fixed singularity if it is a singular point of every solution of the ODE and a movable singularity if the position of the singular point depends on the solution (or equivalently on the initial conditions).

For example, the Ricatti equation$u'' + u^2 = 0$ has a general solution $u(z) = 1/(z-z_0)$, where $z_0\in\mathbb{C}$ is an arbitrary constant. The position of the pole is at $z_0$, hence it depends on a particular solution.

An ODE satisfies the Painlevé property if all solutions are single-valued around every movable singularity.

Martin D. Kruskal, Nalini Joshi, Rod Halburd, Analytic and asymptotic methods for nonlinear singularity analysis: a review and extensions of tests for the Painlevé property, 1996 pdf

Henryk Żołądek, The monodromy group, Monografie Matematyczne 67, 588 pp. Birkhäuser 2006

Vladimir V. Golubev, Lekcii po analitičeskoi teorii differencial’nyh uravnenii. 1ed 1941, 2nd ed. Moscow-Leningrad. 1950; German transl. Vorlesungen ueber Differentialgleichungen im Komplexen, Deutsche Verlag Wissenschaft., 1958

As a possible generalization for PDEs, one can consider analytic movable singularity manifolds as in

J. Weiss, M. Tabor, G. Carnevale, The Painlevé property for partial differential equations, J. Math. Phys. 24, 522–526 (1983)