Painlevé property

The Painlevé property excludes movable critical singularities, such as branch points, of the analytically continued general solution of a complex differential equation. A common stronger formulation requires every movable singularity to be a pole, so local solutions are meromorphic there. Fixed singularities determined by the coefficients are distinguished from movable ones whose locations depend on initial data. Riccati equations with meromorphic coefficients have only movable poles at ordinary points: linearization expresses their solutions as logarithmic derivatives of linear-system solutions. More generally, set $w=f/g$ and solve the holomorphic linear system $f^{\prime }=bf+cg$, $g^{\prime }=-af$. Its quotient satisfies $w^{\prime }=a{w}^2+bw+c$ directly. Initial data $g(z_0)=1$, $f(z_0)=w(z_0)$ recover every local solution, and holomorphic $f,g$ make the quotient a meromorphic function, even at zeros of $a$. At a zero of $g$ with $a\ne 0$, uniqueness prevents $f=0$ there, so $g^{\prime }=-af\ne 0$ and the pole is simple. This proves the absence of movable branching without assuming that the quadratic coefficient never vanishes.