  We consider the problem of constructing solutions to the fractional Yamabe
problem that are singular at a given smooth sub-manifold, and we establish the
classical gluing method of Mazzeo and Pacard for the scalar curvature in the
fractional setting. This proof is based on the analysis of the model linearized
operator, which amounts to the study of an ODE, and thus our main contribution
here is the development of new methods coming from conformal geometry and
scattering theory for the study of non-local ODEs. No traditional phase-plane
analysis is available here. Instead, first, we provide a rigorous construction
of radial fast-decaying solutions by a blow-up argument and a bifurcation
method. Second, we use conformal geometry to rewrite this non-local ODE, giving
a hint of what a non-local phase-plane analysis should be. Third, for the
linear theory, we examine a fractional Schr\"{o}dinger equation with a Hardy
type critical potential. We construct its Green's function, deduce Fredholm
properties, and analyze its asymptotics at the singular points in the spirit of
Frobenius method. Surprisingly enough, a fractional linear ODE may still have a
two-dimensional kernel as in the second order case.
