= Painlevé property
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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 $f'=bf+cg$, $g'=-af$. Its quotient satisfies $w'=aw^2+bw+c$ 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 $a\ne0$, uniqueness prevents $f=0$ there, so $g'=-af\ne0$ and the <pole> is simple. This proves the absence of movable branching without assuming that the quadratic coefficient never vanishes.
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