  Non-local equations cannot be treated using classical ODE theorems.
Nevertheless, several new methods have been introduced in the non-local gluing
scheme of our previous article "On higher dimensional singularities for the
fractional Yamabe problem: a non-local Mazzeo-Pacard program"; we survey and
improve those, and present new applications as well. First, from the explicit
symbol of the conformal fractional Laplacian, a variation of constants formula
is obtained for fractional Hardy operators. We thus develop, in addition to a
suitable extension in the spirit of Caffarelli--Silvestre, an equivalent
formulation as an infinite system of second order constant coefficient ODEs.
Classical ODE quantities like the Hamiltonian and Wro\'nskian may then be
utilized. As applications, we obtain a Frobenius theorem and establish new
Poho\vzaev identities. We also give a detailed proof for the non-degeneracy of
the fast-decay singular solution of the fractional Lane-Emden equation.
