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The problem of Painleve classification of ordinary differential equationslasting since the end of XIX century saw significant advances for the limitedequation order, however not that much for the equations of higher orders. Inthis work we propose the complete Painleve classification for ordinarydifferential equations of the arbitrary order with right-hand side being aquadratic form on the dependent variable and all of its derivatives. The totalof seven classes of the equations with Painleve property have been found. Fiveof them having the order up to four are already known. Sixth one of the otherup to five also appears to be integrable in the known functions. While the onlyseventh class of the unrestricted order appears to be linearizable. Theclassification employs a novel general necessary condition for the Painleveproperty proven in the paper, potentially having a broader application for thePainleve classification of other types of ordinary differential equations.

Painleve Classification of Polynomial Ordinary Differential Equations of Arbitrary Order and Second Degree