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  "body": "\n## Definition\n\nLet $U \\subseteq \\C$ be a domain, and consider an [[ordinary differential equation|0NT64JV96A]]\n$$y^{(n)} = F\\left( z, y, y', \\ldots, y^{(n-1)} \\right),$$\nwhere $F$ is rational in $y, y', \\ldots, y^{(n-1)}$ with coefficients holomorphic in $z \\in U$.\nConsider the singularities of solutions, reached by [[analytic continuation|0M3YQJ81NB]].\nA singularity is *fixed* if its position is determined by the equation alone, such as a singular point of the coefficients.\nOtherwise it is *movable*: its position depends on the initial conditions.\nThe equation has the *Painlevé property* if every movable singularity of every solution is a [[pole|066E7PJCYE]].\n\n## Remarks\n\nFor the first, second and fourth Painlevé equations, the only fixed singular point is $\\infty$, and every solution is a [[meromorphic|0ZZ7ZYDR8W]] function on all of $\\C$.\nFor the other three, solutions can still branch at the fixed singular points, such as $0$, $1$ and $\\infty$ for the sixth equation.\nSome authors only require that no movable singularity is a branch point.\n\nExample: the Riccati equation $y' = y^2$ has the solutions $y = 0$ and $y = \\frac{1}{c - z}$, $c \\in \\C$.\nThe only singularity is the simple pole at $z = c$, which moves with $c$, so this movable singularity is a pole.\nMore generally, every Riccati equation $y' = a(z) + b(z) y + c(z) y^2$ has the Painlevé property.\n\nNon-example: $y' = y^3$ has the solutions $y^2 = \\frac{1}{2 (c - z)}$, so $y = \\pm \\left( 2 (c - z) \\right)^{-1/2}$.\nThe point $z = c$ is a movable branch point and not a pole, so the property fails.\n\nThe first Painlevé equation $y'' = 6 y^2 + z$ has the Painlevé property, and its solutions are not expressible by classical functions in general.\n",
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