Movable singularity

In the theory of ordinary differential equations, a movable singularity is a point where the solution of the equation behaves badly and which is "movable" in the sense that its location depends on the initial conditions of the differential equation. Suppose we have an ordinary differential equation in the complex domain. Any given solution y(x) of this equation may well have singularities at various points (i.e. points at which it is not a regular holomorphic function, such as branch points, essential singularities or poles). A singular point is said to be movable if its location depends on the particular solution we have chosen, rather than being fixed by the equation itself.

For example the equation

has solution ${\displaystyle y={\sqrt {x-c}}}$ for any constant c. This solution has a branchpoint at ${\displaystyle x=c}$, and so the equation has a movable branchpoint (since it depends on the choice of the solution, i.e. the choice of the constant c).

...
Wikipedia