 Movable singularity of a complex differential equation

A solution has a movable singularity if its location varies with the initial data. For example $w^{\prime }=w^2$ has solutions $w=-1/(z-c)$, whose pole moves with $c$. The Painlevé determinateness theorem restricts movable singularities of first-order equations rational in the dependent variable to poles and algebraic branch points.