Global branching of solutions to ODEs and integrability

We consider a natural generalisation of the Painlev\'e property and use it to identify the known integrable cases of the Lane-Emden equation with a real positive index. We classify certain first-order ordinary differential equations with this property and find necessary conditions for a large family of second-order equations. We consider ODEs such that, given any simply connected domain $\Omega$ not containing fixed singularities of the equation, the Riemann surface of any solution obtained by analytic continuation along curves in $\Omega$ has a finite number of sheets over $\Omega$.