  The problem of Painleve classification of ordinary differential equations
lasting since the end of XIX century saw significant advances for the limited
equation order, however not that much for the equations of higher orders. In
this work we propose the complete Painleve classification for ordinary
differential equations of the arbitrary order with right-hand side being a
quadratic form on the dependent variable and all of its derivatives. The total
of seven classes of the equations with Painleve property have been found. Five
of them having the order up to four are already known. Sixth one of the other
up to five also appears to be integrable in the known functions. While the only
seventh class of the unrestricted order appears to be linearizable. The
classification employs a novel general necessary condition for the Painleve
property proven in the paper, potentially having a broader application for the
Painleve classification of other types of ordinary differential equations.
